**************************** CHAPTER 2 ********************************* MTB > exec 'craps' MTB > ################################################################## MTB > # MACRO 'CRAPS' # MTB > # -------------------------------------------------------------- # MTB > # PLAYS GAME OF CRAPS # MTB > ################################################################## HOW MANY GAMES DO YOU WANT TO PLAY? DATA> 2 NOTE: Type 'y' and return to play: y GAME 1 ROLLS 6 3 8 7 RESULT You lose! NOTE: Type 'y' and return to play: y GAME 2 ROLLS 8 6 4 9 9 7 RESULT You lose! Type 'y' and return for some summary statistics: y ROWS: 1ST_ROLL COLUMNS: WIN? 0 ALL 6 1 1 8 1 1 ALL 2 2 CELL CONTENTS -- COUNT ROWS: N_ROLLS COLUMNS: WIN? 0 ALL 4 1 1 6 1 1 ALL 2 2 CELL CONTENTS -- COUNT MTB > exec 'yahtzee' MTB > ################################################################## MTB > # MACRO 'YAHTZEE' # MTB > # -------------------------------------------------------------- # MTB > # PLAYS GAME OF YAHTZEE # MTB > # PLAYER CAN DECIDE WHICH DICE TO DISCARD # MTB > ################################################################## Type 'y' and return to start rolling dice: y ######################### # ROLL 1 # ######################### rolls 4 6 5 3 4 result small straight ENTER (USING SEQUENCES OF 0'S AND 1'S) WHICH DICE TO KEEP: DATA> 1 0 0 0 1 ######################### # ROLL 2 # ######################### rolls 4 4 1 4 4 result 4 of a kind ENTER (USING SEQUENCES OF 0'S AND 1'S) WHICH DICE TO KEEP: DATA> 1 1 0 1 1 ######################### # ROLL 3 # ######################### rolls 4 4 5 4 4 result 4 of a kind MTB > exec 'yahtz_au' MTB > ################################################################## MTB > # MACRO 'YAHTZ_AU' # MTB > # -------------------------------------------------------------- # MTB > # PLAYS GAME OF YAHTZEE # MTB > # COMPUTER DECIDES ON WHICH DICE TO DISCARD # MTB > ################################################################## Type 'y' and return to start rolling dice: y ################################# # 1ST ROLL # ################################# rolls 3 5 1 2 4 result large straight keep 1 1 1 1 1 TYPE 'y' AND RETURN FOR NEXT ROLL: y ################################# # 2ND ROLL # ################################# rolls 3 5 1 2 4 result large straight keep 1 1 1 1 1 TYPE 'y' AND RETURN FOR NEXT ROLL: y ################################# # 3RD ROLL # ################################# rolls 3 5 1 2 4 result large straight MTB > exec 'yahtz_re' 9 out 2 2 5 out 3 3 3 out 5 0 0 out 2 3 3 out 4 4 8 out 3 3 3 out 2 4 5 out 2 3 3 out 2 3 3 MTB > exec 'bball' MTB > MTB > ################################################################## MTB > # MACRO 'BBALL' # MTB > # -------------------------------------------------------------- # MTB > # SIMULATES A SEASON OF BASEBALL # MTB > ################################################################## INPUT TEAM STRENGTHS: DATA> -1 -.5 0 .5 1 DATA> end ROWS: winner COLUMNS: loser 1 2 3 4 5 ALL 1 0 4 3 3 0 10 2 4 0 4 4 1 13 3 5 4 0 5 2 16 4 5 4 3 0 4 16 5 8 7 6 4 0 25 ALL 22 19 16 16 7 80 CELL CONTENTS -- COUNT winner Count 1 10 2 13 3 16 4 16 5 25 N= 80 MTB > exec 'bball_re' 9 winner Count 1 6 2 11 3 12 4 24 5 27 N= 80 winner Count 1 6 2 11 3 13 4 26 5 24 N= 80 winner Count 1 11 2 9 3 13 4 20 5 27 N= 80 winner Count 1 11 2 12 3 15 4 18 5 24 N= 80 winner Count 1 8 2 6 3 15 4 22 5 29 N= 80 winner Count 1 12 2 10 3 15 4 20 5 23 N= 80 winner Count 1 8 2 13 3 16 4 16 5 27 N= 80 winner Count 1 7 2 16 3 13 4 25 5 19 N= 80 winner Count 1 11 2 15 3 11 4 16 5 27 N= 80 MTB > prin 'nwins_1'-'nwins_5' Row nwins_1 nwins_2 nwins_3 nwins_4 nwins_5 1 10 13 16 16 25 2 6 11 12 24 27 3 6 11 13 26 24 4 11 9 13 20 27 5 11 12 15 18 24 6 8 6 15 22 29 7 12 10 15 20 23 8 8 13 16 16 27 9 7 16 13 25 19 10 11 15 11 16 27 MTB > prin 'place_1'-'place_5' Row place_1 place_2 place_3 place_4 place_5 1 5.0 4 2.5 2.5 1 2 5.0 4 3.0 2.0 1 3 5.0 4 3.0 1.0 2 4 4.0 5 3.0 2.0 1 5 5.0 4 3.0 2.0 1 6 4.0 5 3.0 2.0 1 7 4.0 5 3.0 2.0 1 8 5.0 4 2.5 2.5 1 9 5.0 3 4.0 1.0 2 10 4.5 3 4.5 2.0 1 **************************** CHAPTER 3 ********************************* MTB > MTB > exec 'bayes_se' MTB > ################################################################## MTB > # MACRO 'BAYES_SE' # MTB > # -------------------------------------------------------------- # MTB > # BAYES RULE FOR A FINITE NUMBER OF MODELS # MTB > # AND FINITE NUMBER OF OUTCOMES. # MTB > # # MTB > # THIS PROGRAM SETS UP MODELS, PRIOR, AND LIKELIHOODS. # MTB > # # MTB > # THE MACRO 'BAYES' IMPLEMENTS BAYES RULE FOR A SEQUENCE # MTB > # OF INDEPENDENT OUTCOMES. # MTB > ################################################################## INPUT NUMBER OF MODELS: DATA> 2 INPUT NAMES OF MODELS (ONE NAME ON EACH LINE): DATA> have disease DATA> don't have disease INPUT PRIOR PROBABILITIES OF MODELS: DATA> .001 .999 INPUT THE NUMBER OF POSSIBLE OUTCOMES: DATA> 2 INPUT THE NAME OF EACH OBSERVATION: (ONE OBSERVATION ON A LINE) DATA> + DATA> - INPUT LIKELIHOODS OF EACH MODEL: MODEL 1 DATA> .95 .05 1 rows read. MODEL 2 DATA> .05 .95 1 rows read. OBSERVATION NAMES: Row OBS OBS_NAME 1 OUT_1 + 2 OUT_2 - TABLE OF PROBABILITIES OF MODELS AND OUTCOMES: Row MODEL NAME PRIOR OUT_1 OUT_2 1 1 have disease 0.001 0.95 0.05 2 2 don't have d 0.999 0.05 0.95 MTB > exec 'bayes' MTB > ################################################################## MTB > # MACRO 'BAYES' # MTB > # -------------------------------------------------------------- # MTB > # IMPLEMENTS BAYES RULE FOR A SEQUENCE OF INDEPENDENT OUTCOMES. # MTB > # THE PROGRAM 'BAYES_SETUP' MUST BE RUN FIRST TO # MTB > # SET UP MODELS, PRIOR, AND LIKELIHOODS. # MTB > ################################################################## INPUT NUMBER OF OBSERVATIONS: DATA> 1 INPUT OBSERVATIONS: (ONE OBSERVATION NAME ON A LINE:) DATA> + OUTCOME + Row MODEL NAME PRIOR LIKE PRODUCT POST 1 1 have disease 0.001 0.95 0.00095 0.018664 2 2 don't have d 0.999 0.05 0.04995 0.981336 SUMMARY OF PRIOR AND POSTERIOR MODEL PROBABILITIES: Row OBS_NO OUTCOMES PROB_M1 PROB_M2 1 0 0.0010000 0.999000 2 1 + 0.0186640 0.981336 **************************** CHAPTER 4 ********************************* MTB > MTB > name c1 'p' c2 'prior' MTB > set c1 DATA> .2 .25 .3 .35 DATA> end MTB > set 'prior' DATA> .25 .25 .25 .25 DATA> end MTB > exec 'p_disc' MTB > ################################################################## MTB > # MACRO 'P_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A BINOMIAL PROPORTION P USING # MTB > # A FINITE COLLECTION OF P MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF P IN COLUMN 'P' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES: DATA> 5 15 Row p prior P_x_PRIO LIKE PRODUCT POST P_x_POST 1 0.20 0.25 0.0500 862742 215686 0.255595 0.051119 2 0.25 0.25 0.0625 1000000 250000 0.296259 0.074065 3 0.30 0.25 0.0750 884011 221003 0.261897 0.078569 4 0.35 0.25 0.0875 628668 157167 0.186249 0.065187 5 0.2750 0.268940 - 0.30+ * - * PROB - * 2 * - 2 * 2 - * * * 0.20+ 2 2 2 - * * 2 2 - 2 * * 2 - * 2 2 3 - 2 * * 2 0.10+ * * 2 2 - 2 2 * 2 - 2 * 2 2 - * * * 2 - 2 2 2 2 0.00+ * * * 2 ------+---------+---------+---------+---------+---------+MODEL 0.210 0.240 0.270 0.300 0.330 0.360 MTB > set c1 DATA> .2:.5/.01 DATA> end MTB > let c2=1+0*c1 MTB > exec 'p_disc' MTB > ################################################################## MTB > # MACRO 'P_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A BINOMIAL PROPORTION P USING # MTB > # A FINITE COLLECTION OF P MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF P IN COLUMN 'P' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES: DATA> 132 236 Row p prior P_x_PRIO LIKE PRODUCT POST P_x_POST 1 0.20 0.0322581 0.006452 0 0.0 0.000000 0.000000 2 0.21 0.0322581 0.006774 0 0.0 0.000000 0.000000 3 0.22 0.0322581 0.007097 0 0.0 0.000000 0.000000 4 0.23 0.0322581 0.007419 0 0.0 0.000000 0.000000 5 0.24 0.0322581 0.007742 2 0.1 0.000000 0.000000 6 0.25 0.0322581 0.008065 22 0.7 0.000004 0.000001 7 0.26 0.0322581 0.008387 168 5.4 0.000027 0.000007 8 0.27 0.0322581 0.008710 986 31.8 0.000157 0.000042 9 0.28 0.0322581 0.009032 4623 149.1 0.000738 0.000207 10 0.29 0.0322581 0.009355 17503 564.6 0.002795 0.000810 11 0.30 0.0322581 0.009677 54043 1743.3 0.008628 0.002589 12 0.31 0.0322581 0.010000 137316 4429.5 0.021924 0.006796 13 0.32 0.0322581 0.010323 289396 9335.4 0.046205 0.014786 14 0.33 0.0322581 0.010645 509435 16433.4 0.081337 0.026841 15 0.34 0.0322581 0.010968 753677 24312.2 0.120333 0.040913 16 0.35 0.0322581 0.011290 942155 30392.1 0.150426 0.052649 17 0.36 0.0322581 0.011613 1000000 32258.1 0.159661 0.057478 18 0.37 0.0322581 0.011935 904873 29189.5 0.144473 0.053455 19 0.38 0.0322581 0.012258 700599 22600.0 0.111858 0.042506 20 0.39 0.0322581 0.012581 465547 15017.7 0.074330 0.028989 21 0.40 0.0322581 0.012903 266206 8587.3 0.042503 0.017001 22 0.41 0.0322581 0.013226 131264 4234.3 0.020958 0.008593 23 0.42 0.0322581 0.013548 55913 1803.6 0.008927 0.003749 24 0.43 0.0322581 0.013871 20601 664.6 0.003289 0.001414 25 0.44 0.0322581 0.014194 6572 212.0 0.001049 0.000462 26 0.45 0.0322581 0.014516 1817 58.6 0.000290 0.000131 27 0.46 0.0322581 0.014839 435 14.0 0.000069 0.000032 28 0.47 0.0322581 0.015161 90 2.9 0.000014 0.000007 29 0.48 0.0322581 0.015484 16 0.5 0.000003 0.000001 30 0.49 0.0322581 0.015806 3 0.1 0.000000 0.000000 31 0.50 0.0322581 0.016129 0 0.0 0.000000 0.000000 32 0.350000 0.359460 - * 0.150+ * * - 2 2 2 PROB - * * * - ** * 2 - 22 * *2 0.100+ 2* 2 *2 - ** * 2* - 2 22 * *2 - 3 2* * 22 3 - 2 ** 2 *2 3 0.050+ * 2 22 * ** 2 - 4 3 2* * 22 3 4 - 5 2 ** * *2 3 5 - 74 3 22 2 *2 2 46 - 9 94 2 2* * 22 3 5+ 9 0.000+ + + ++ + ++ + ++ + 53 2 ** * ** 2 35 + ++ + ++ + ++ +---------+---------+---------+---------+---------+------MODEL 0.180 0.240 0.300 0.360 0.420 0.480 MTB > exec 'disc_sum' INPUT NUMBER OF COLUMN WHICH CONTAINS VALUES OF VARIABLE: DATA> 1 INPUT NUMBER OF COLUMN WHICH CONTAINS PROBABILITIES: DATA> 51 TYPE 'y' TO SEE A PLOT OF THE PROBABILITIES: n TYPE 'y' TO GET SUMMARIES OF THE DISTRIBUTION: y Row MODE MEAN STD 1 0.36 0.359460 0.0249122 TYPE 'y' TO COMPUTE CUMULATIVE PROBABILITIES: -------------------------------------------------------------------- Input values of variable of interest. The output is the column of values and the column of cumulative probabilities PROB_LE. -------------------------------------------------------------------- y DATA> .2 .39 DATA> end Row VALUE PROB_LE 1 0.20 0.000000 2 0.39 0.922897 TYPE 'y' TO COMPUTE PROBABILITY INTERVALS: -------------------------------------------------------------------- Input list of probabilities. For each probability p, the set of values of the variable for which the probability content of the set exceeds p is given. -------------------------------------------------------------------- y DATA> .5 .9 DATA> end PROB_SET 0.574893 SET 0.34 0.35 0.36 0.37 -------------------------------------------------------------------------- PROB_SET 0.931126 SET 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 -------------------------------------------------------------------------- MTB > set 'p' DATA> .2 .25 .3 .35 DATA> end MTB > set 'prior' DATA> .25 .25 .25 .25 DATA> end MTB > exec 'p_disc_p' MTB > ################################################################## MTB > # MACRO 'P_DISC_P' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # PREDICTIVE INFERENCE FOR BINOMIAL SAMPLING # MTB > # AND FINITE COLLECTION OF P MODELS. # MTB > # -------------------------------------------------------------- # MTB > # INPUT: MODELS IN 'P', PROBABILITIES IN 'PRIOR' OR 'POST' # MTB > # OUTPUT: NUMBER OF SUCCESSES IN COLUMN 'SUCC' AND # MTB > # PREDICTIVE PROBABILITIES IN COLUMN 'PRED' # MTB > ################################################################## INPUT 1 IF PROBABILITIES ARE IN 'PRIOR' OR 2 IF PROBABILITIES ARE IN 'POST': DATA> 1 INPUT NUMBER OF TRIALS: DATA> 20 INPUT RANGE (LOW AND HIGH VALUES) FOR NUMBER OF SUCCESSES: DATA> 0 20 PREDICTIVE DISTRIBUTION OF NUMBER OF SUCCESSES: Row SUCC PRED 1 0 0.003920 2 1 0.021895 3 2 0.060422 4 3 0.110780 5 4 0.153032 6 5 0.170739 7 6 0.160145 8 7 0.128905 9 8 0.089699 10 9 0.053915 11 10 0.027846 12 11 0.012266 13 12 0.004565 14 13 0.001420 15 14 0.000364 16 15 0.000075 17 16 0.000012 18 17 0.000002 19 18 0.000000 20 19 0.000000 21 20 0.000000 - 0.180+ - 2 PROB - * * 2 - * * * - 2 2 2 * 0.120+ 2 * * 2 - 2 * 2 2 2 - 2 2 * * 2 - 2 * * 2 * 3 - 3 2 2 * 2 3 0.060+ 3 2 * * 2 2 2 - 4 2 2 2 * 2 3 5 - 4 2 2 * 2 2 3 4 - 4 4 2 * * * 2 2 5 8 - + 4 2 2 2 2 2 3 4 8 + 0.000+ + 6 2 2 * * * * 2 3 5 + + + + + + + + + + +---------+---------+---------+---------+---------+------S 0.0 4.0 8.0 12.0 16.0 20.0 MTB > exec 'beta_sel' What is the probability of a success on the first trial? DATA> .3 If the first trial is a success, what is the conditional probability of a success on the second trial? (This should be larger than the first number you gave.) DATA> .32 The matching values of the beta parameters a and b corresponding to your predictive probabilities are given by: Row a b 1 10.200 23.8 MTB > exec 'p_beta' MTB > ################################################################## MTB > # MACRO 'P_BETA' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # FOR A BETA(A,B) DISTRIBUTION, GRAPHS DENSITY CURVE, # MTB > # AND COMPUTES CUMULATIVE PROBABILITIES AND PERCENTILES. # MTB > ################################################################## INPUT VALUES OF BETA PARAMETERS A AND B: DATA> 10.2 23.8 TYPE 'y' TO SEE A PLOT OF THE BETA DENSITY: y - 2 4.8+ * - * * density - - * * - 3.2+ * - * - * - * - * 1.6+ - * * - * - * * - * ** 0.0+ ****** * *2 ******************** +---------+---------+---------+---------+---------+------p 0.00 0.20 0.40 0.60 0.80 1.00 TYPE 'y' TO COMPUTE CUMULATIVE PROBABILITIES: ---------------------------------------------------------------- Input values of P of interest. The output is the column of values P and the column of cumulative probabilities PROB_LT. ---------------------------------------------------------------- y DATA> .1 .2 .3 .4 .5 DATA> end Row p PROB_LT 1 0.1 0.000793 2 0.2 0.092903 3 0.3 0.519992 4 0.4 0.894790 5 0.5 0.991727 TYPE 'y' TO COMPUTE PERCENTILES: ---------------------------------------------------------------- Input probabilities for which you wish to compute percentiles. The output is the probabilities in the column PROB and the corresponding percentiles in the column PERCNTLE. ---------------------------------------------------------------- y DATA> .05 .25 .5 .75 .95 DATA> end Row PROB PERCNTLE 1 0.05 0.179486 2 0.25 0.244887 3 0.50 0.296039 4 0.75 0.350879 5 0.95 0.434082 MTB > erase c1-c100 MTB > exec 'p_beta_p' MTB > ################################################################## MTB > # MACRO 'P_BETA_P' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # PREDICTIVE INFERENCE FOR BINOMIAL SAMPLING # MTB > # AND A BETA PRIOR FOR P # MTB > # -------------------------------------------------------------- # MTB > # OUTPUT: NUMBER OF SUCCESSES IN COLUMN 'SUCC' AND # MTB > # PREDICTIVE PROBABILITIES IN COLUMN 'PRED' # MTB > ################################################################## INPUT VALUES OF BETA PARAMETERS A AND B: DATA> 20.4 47.6 INPUT NUMBER OF TRIALS: DATA> 20 INPUT RANGE (LOW AND HIGH VALUES) FOR NUMBER OF SUCCESSES: DATA> 0 20 PREDICTIVE DISTRIBUTION OF NUMBER OF SUCCESSES: Row SUCC PRED 1 0 0.002119 2 1 0.012982 3 2 0.040232 4 3 0.083703 5 4 0.130884 6 5 0.163249 7 6 0.168285 8 7 0.146624 9 8 0.109537 10 9 0.070782 11 10 0.039741 12 11 0.019405 13 12 0.008219 14 13 0.003001 15 14 0.000935 16 15 0.000245 17 16 0.000052 18 17 0.000009 19 18 0.000001 20 19 0.000000 21 20 0.000000 - 0.180+ - * * PROB - * 2 - 2 * 2 - * * 2 * 0.120+ 2 2 * 2 - 2 * * 2 2 - 2 * 2 * 2 - 2 2 2 * 2 2 - 3 * * 2 * 2 2 0.060+ 3 2 2 * 2 3 3 - 2 2 * 2 2 2 4 - 6 3 2 2 * * 2 3 5 - 6 3 2 * * 2 2 3 6 2 - + 6 3 2 2 2 2 2 4 6 + 6 0.000+ + + 3 2 * * * * 2 2 4 7 + + + + + + + + + +---------+---------+---------+---------+---------+------S 0.0 4.0 8.0 12.0 16.0 20.0 MTB > exec 'p_beta_t' MTB > ################################################################## MTB > # MACRO 'P_BETA_T' # MTB > # -------------------------------------------------------------- # MTB > # TESTS THE HYPOTHESIS THAT P = P0 USING A BETA PRIOR # MTB > ################################################################## ENTER THE NULL HYPOTHESIS PROPORTION P0: DATA> .5 ENTER THE PRIOR PROBABILITY OF P0: DATA> .5 FOR THE ALTERNATIVE HYPOTHESIS THAT P = P0, ENTER THE NUMBERS A AND B OF THE BETA(A, B) DISTRIBUTION: DATA> 10 10 ENTER THE OBSERVED NUMBER OF SUCCESSES AND FAILURES: DATA> 22 28 The Bayes factor in favor of the null hypothesis is: BF_HK 1.45372 The Bayes factor against the null hypothesis is: BF_KH 0.68789 The posterior probability of the null hypothesis is: prob_H 0.592456 DATA> 0 MTB > exec 'p_hist_p' MTB > ################################################################## MTB > # MACRO 'P_HIST_P' # MTB > # -------------------------------------------------------------- # MTB > # LEARNING ABOUT A BINOMIAL PROPORTION P # MTB > # USING A HISTOGRAM PRIOR. # MTB > ################################################################## INPUT INTERVAL MIDPOINTS: DATA> .225 .275 .325 .375 DATA> end INPUT PRIOR PROBABILITIES OF INTERVALS: DATA> .4 .4 .15 .05 INPUT NUMBER OF SIMULATED VALUES: DATA> 1000 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES: DATA> 4 36 The prior and posterior probabilities of the intervals: Row MIDS LO HI PRIOR POST 1 0.225 0.20 0.25 0.40 0.821976 2 0.275 0.25 0.30 0.40 0.165562 3 0.325 0.30 0.35 0.15 0.012053 4 0.375 0.35 0.40 0.05 0.000408 **************************** CHAPTER 5 ********************************* MTB > exec 'pp_disc' MTB > ################################################################## MTB > # MACRO 'PP_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT 2 BINOMIAL PROPORTIONS # MTB > # USING A FINITE COLLECTION OF P1,P2 MODELS - UNIFORM PRIOR # MTB > # -------------------------------------------------------------- # MTB > # OUTPUT: COLUMNS 'P1', 'P2', 'PRIOR', 'POST', 'DIFF', 'P_DIFF'# MTB > ################################################################## FOR EACH P DISTRIBUTION: ------------------------ INPUT LO AND HI VALUES: DATA> 0 1 INPUT NUMBER OF MODELS: DATA> 11 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN FIRST SAMPLE: DATA> 2 13 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN SECOND SAMPLE: DATA> 14 1 Posterior distribution of P1 and P2: (Rows and columns are expressed in percentage format.) ROWS: PER_1 COLUMNS: PER_2 0 10 20 30 40 50 60 70 0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000021 0.000389 0.003996 0.025940 20 0.000000 0.000000 0.000000 0.000000 0.000018 0.000337 0.003457 0.022441 30 0.000000 0.000000 0.000000 0.000000 0.000007 0.000133 0.001371 0.008899 40 0.000000 0.000000 0.000000 0.000000 0.000002 0.000032 0.000329 0.002133 50 0.000000 0.000000 0.000000 0.000000 0.000000 0.000005 0.000048 0.000311 60 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000004 0.000025 70 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001 80 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 90 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1000.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 80 90 100 0 0.000000 0.000000 0.000000 10 0.112142 0.291657 0.000000 20 0.097016 0.252319 0.000000 30 0.038471 0.100055 0.000000 40 0.009219 0.023978 0.000000 50 0.001346 0.003502 0.000000 60 0.000107 0.000277 0.000000 70 0.000003 0.000009 0.000000 80 0.000000 0.000000 0.000000 90 0.000000 0.000000 0.000000 1000.000000 0.000000 0.000000 CELL CONTENTS -- POST:DATA TYPE 'Y' AND RETURN TO SEE A GRAPH OF THE POSTERIOR DISTRIBUTION: y - 1.05+ - . . . . . . . . . . . P1 - . . . . . . . . . . . - - . . . . . . . . . . . 0.70+ . . . . . . . . . . . - . . . . . . . . . . . - - . . . . . . . . . . . - . . . . . . . . / / . 0.35+ - . . . . . . . / / X . - . . . . . . . / X X . - - . . . . . . . / X X . 0.00+ . . . . . . . . . . . +---------+---------+---------+---------+---------+------P2 0.00 0.20 0.40 0.60 0.80 1.00 '0' < -3.0E-02 < '.' < 8.26E-03 < '/' < 4.66E-02 < 'X' TYPE 'Y' AND RETURN TO SEE A TABLE OF THE POSTERIOR DISTRIBUTION OF THE DIFFERENCE IN PROBABILITIES P2-P1: y Row DIFF P_DIFF 1 -1.0 0.000000 2 -0.9 0.000000 3 -0.8 0.000000 4 -0.7 0.000000 5 -0.6 0.000000 6 -0.5 0.000000 7 -0.4 0.000000 8 -0.3 0.000000 9 -0.2 0.000000 10 -0.1 0.000001 11 0.0 0.000011 12 0.1 0.000116 13 0.2 0.000907 14 0.3 0.005484 15 0.4 0.025466 16 0.5 0.088886 17 0.6 0.223011 18 0.7 0.364461 19 0.8 0.291657 20 0.9 0.000000 21 1.0 0.000000 MTB > exec 'pp_disct' FOR EACH P DISTRIBUTION: ------------------------ INPUT LO AND HI VALUES: DATA> 0 1 INPUT NUMBER OF MODELS: DATA> 11 INPUT PROBABILITY THAT P1=P2: DATA> .5 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN FIRST SAMPLE: DATA> 2 13 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN SECOND SAMPLE: DATA> 14 1 Posterior distribution of P1 and P2: (Rows and columns are expressed in percentage format.) ROWS: PER_1 COLUMNS: PER_2 0 10 20 30 40 50 60 70 0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000021 0.000389 0.003996 0.025937 20 0.000000 0.000000 0.000000 0.000000 0.000018 0.000337 0.003457 0.022439 30 0.000000 0.000000 0.000000 0.000001 0.000007 0.000133 0.001371 0.008898 40 0.000000 0.000000 0.000000 0.000000 0.000017 0.000032 0.000329 0.002132 50 0.000000 0.000000 0.000000 0.000000 0.000000 0.000047 0.000048 0.000311 60 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000038 0.000025 70 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000008 80 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 90 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1000.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 80 90 100 0 0.000000 0.000000 0.000000 10 0.112131 0.291628 0.000000 20 0.097007 0.252294 0.000000 30 0.038467 0.100045 0.000000 40 0.009218 0.023975 0.000000 50 0.001346 0.003501 0.000000 60 0.000107 0.000277 0.000000 70 0.000003 0.000009 0.000000 80 0.000000 0.000000 0.000000 90 0.000000 0.000000 0.000000 1000.000000 0.000000 0.000000 CELL CONTENTS -- POST:DATA TYPE 'Y' AND RETURN TO SEE A GRAPH OF THE POSTERIOR DISTRIBUTION: y - 1.05+ - . . . . . . . . . . . P1 - . . . . . . . . . . . - - . . . . . . . . . . . 0.70+ . . . . . . . . . . . - . . . . . . . . . . . - - . . . . . . . . . . . - . . . . . . . . / / . 0.35+ - . . . . . . . / / X . - . . . . . . . / X X . - - . . . . . . . / X X . 0.00+ . . . . . . . . . . . +---------+---------+---------+---------+---------+------P2 0.00 0.20 0.40 0.60 0.80 1.00 '0' < -3.0E-02 < '.' < 8.26E-03 < '/' < 4.65E-02 < 'X' TYPE 'Y' AND RETURN TO SEE A TABLE OF THE POSTERIOR DISTRIBUTION OF THE DIFFERENCE IN PROBABILITIES P2-P1: y Row DIFF P_DIFF 1 -1.0 0.000000 2 -0.9 0.000000 3 -0.8 0.000000 4 -0.7 0.000000 5 -0.6 0.000000 6 -0.5 0.000000 7 -0.4 0.000000 8 -0.3 0.000000 9 -0.2 0.000000 10 -0.1 0.000001 11 0.0 0.000111 12 0.1 0.000116 13 0.2 0.000907 14 0.3 0.005484 15 0.4 0.025464 16 0.5 0.088877 17 0.6 0.222989 18 0.7 0.364424 19 0.8 0.291628 20 0.9 0.000000 21 1.0 0.000000 MTB > read c3-c11 DATA> 0 0 0 .02 .02 .02 .06 .120 .16 DATA> 0 0 0 .0150 .015 .015 .045 .09 .12 DATA> 0 0 0 .0075 .0075 .0075 .0225 .045 .06 DATA> 0 0 0 .0025 .0025 .0025 .0075 .015 .02 DATA> 0 0 0 .0025 .0025 .0025 .0075 .015 .02 DATA> 0 0 0 .0025 .0025 .0025 .0075 .015 .02 DATA> 0 0 0 0 0 0 0 0 0 DATA> 0 0 0 0 0 0 0 0 0 DATA> 0 0 0 0 0 0 0 0 0 DATA> end MTB > exec 'pp_discm' INPUT THE NUMBER OF THE COLUMN WHICH CONTAINS THE P1 VALUES: DATA> 1 INPUT THE NUMBER OF THE COLUMN WHICH CONTAINS THE P2 VALUES: DATA> 2 INPUT THE NUMBER OF THE FIRST COLUMN WHICH CONTAINS THE PROBABILITIES: DATA> 3 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN FIRST SAMPLE: DATA> 2 13 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN SECOND SAMPLE: DATA> 14 1 Posterior distribution of P1 and P2: (Rows and columns are expressed in percentage format.) ROWS: PER_1 COLUMNS: PER_2 10 20 30 40 50 60 70 80 10 0.000000 0.000000 0.000000 0.000004 0.000070 0.000723 0.014082 0.121756 20 0.000000 0.000000 0.000000 0.000002 0.000046 0.000469 0.009137 0.079000 30 0.000000 0.000000 0.000000 0.000000 0.000009 0.000093 0.001812 0.015663 40 0.000000 0.000000 0.000000 0.000000 0.000001 0.000007 0.000145 0.001251 50 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001 0.000021 0.000183 60 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000002 0.000014 70 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 80 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 90 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 90 10 0.422216 20 0.273952 30 0.054316 40 0.004339 50 0.000634 60 0.000050 70 0.000000 80 0.000000 90 0.000000 CELL CONTENTS -- POST:DATA TYPE 'Y' AND RETURN TO SEE A GRAPH OF THE POSTERIOR DISTRIBUTION: y P1 - . . . . . . . . . - - . . . . . . . . . 0.75+ - . . . . . . . . . - - . . . . . . . . . - 0.50+ . . . . . . . . . - - . . . . . . . . . - - . . . . . . . / / 0.25+ - . . . . . . . X X - - . . . . . . / X X ------+---------+---------+---------+---------+---------+P2 0.15 0.30 0.45 0.60 0.75 0.90 '0' < -4.5E-02 < '.' < 1.23E-02 < '/' < 7.00E-02 < 'X' TYPE 'Y' AND RETURN TO SEE A TABLE OF THE POSTERIOR DISTRIBUTION OF THE DIFFERENCE IN PROBABILITIES P2-P1: y Row DIFF P_DIFF 1 -0.8 0.000000 2 -0.7 0.000000 3 -0.6 0.000000 4 -0.5 0.000000 5 -0.4 0.000000 6 -0.3 0.000000 7 -0.2 0.000000 8 -0.1 0.000000 9 0.0 0.000000 10 0.1 0.000004 11 0.2 0.000054 12 0.3 0.000520 13 0.4 0.004236 14 0.5 0.029862 15 0.6 0.147399 16 0.7 0.395708 17 0.8 0.422216 MTB > exec 'pp_beta' MTB > ################################################################## MTB > # MACRO 'PP_BETA' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT 2 BINOMIAL PROPORTIONS # MTB > # USING INDEPENDENT BETA PRIORS FOR P1 AND P2 AND SIMULATION. # MTB > ################################################################## FOR PROPORTION P1, ENTER VALUES OF BETA PARAMETERS A1 AND B1: DATA> 3 14 FOR PROPORTION P2, ENTER VALUES OF BETA PARAMETERS A2 AND B2: DATA> 15 2 HOW MANY VALUES OF (P1, P2) DO YOU WISH TO SIMULATE? DATA> 1000 TYPE 'y' TO SEE A PLOT OF THE JOINT DISTRIBUTION OF P1 AND P2: y 1.05+ - 596+++++++++57765... P2 - 5++++++++++++++755352... . - .5++++++++++5++322.2.. .. - .4+448+++8+83352. . . 0.70+ 23437.....32. - . .33 . 3. . . - . . 2 - . - 0.35+ - - - - 0.00+ +---------+---------+---------+---------+---------+P1 0.00 0.20 0.40 0.60 0.80 1.00 TYPE 'y' TO SEE A PLOT OF THE DISTRIBUTION OF THE DIFFERENCE IN PROPORTIONS P2-P1: y Each dot represents 4 points . : : . :. . . ::.:::: : : :::::::::::.: : :::::::::::::.:: : ..:::::::::::::::::::.::: . :::::::::::::::::::::::::::. . . ..: .:.::.:::::::::::::::::::::::::::::.... -------+---------+---------+---------+---------+---------P2-P1 0.30 0.45 0.60 0.75 0.90 1.05 TYPE 'y' to COMPUTE PROBABILITIES OF IMPROVEMENT FOR P2-P1: ----------------------------------------------------------- Input values of possible improvement. The output is the probabilty PdALx that P2-P1 exceeds each improvement value x. The column sim_se gives simulation standard errors for the estimated probabilities. ------------------------------------------------------------ y Executing from file: MINITAB.MTB DATA> .5 .6 .7 .8 .9 DATA> 0 DATA> end Row x PdALx sim_se 1 0.5 0.956 0.006 2 0.6 0.830 0.012 3 0.7 0.546 0.016 4 0.8 0.229 0.013 5 0.9 0.029 0.005 6 0.0 1.000 0.000 MTB > exec 'pp_bet_t' MTB > ################################################################## MTB > # MACRO 'PP_BET_T' # MTB > # -------------------------------------------------------------- # MTB > # TEST IF 2 BINOMIAL PROPORTIONS ARE EQUAL # MTB > # USING CONTINUOUS P1,P2 MODELS (BETA PRIORS). # MTB > ################################################################## Enter the prior probability of the null hypothesis H of equality: DATA> .5 UNDER THE NULL HYPOTHESIS H THAT P1=P2 --------------------------------------- Enter the numbers a and b of the beta(a, b) distribution: DATA> 1 1 UNDER THE ALTERNATIVE HYPOTHESIS K THAT P1=P2 ---------------------------------------------- Enter the numbers a1 and b1 of the beta(a1, b1) distribution on P1: DATA> 1 1 Enter the numbers a2 and b2 of the beta(a2, b2) distribution on P2: DATA> 1 1 THE DATA --------- Enter the number of observed successes and failures for the 1st sample: DATA> 2 13 Enter the number of observed successes and failures for the 2nd sample: DATA> 14 1 The Bayes factor in favor of the null hypothesis is: BF_HK 0.0000894 The Bayes factor against the null hypothesis is: BF_KH 11180.8 The posterior probability of the null hypothesis is: prob_H 0.0000894 MTB > exec 'pp_exch' MTB > ################################################################## MTB > # MACRO 'PP_EXCH' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT 2 BINOMIAL PROPORTIONS # MTB > # USING AN EXCHANGEABLE PRIOR FOR P1 AND P2. # MTB > ################################################################## INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN 1ST GROUP: DATA> 0 0 INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN 2ND GROUP: DATA> 0 0 INPUT STANDARD DEVIATION OF THE LOGITS T1 AND T2: DATA> .5 INPUT NUMBER OF SIMULATED VALUES: DATA> 1000 TYPE 'y' AND RETURN TO SEE A PLOT OF THE JOINT DISTRIBUTION OF P1 AND P2: y 1.05+ - ...5 post_p2 - 2. 4.22.2.32.435 . - . .. 2 5.3.23.2622.39. - 3 .3 22 7546.3.+59.25 2. 0.70+ .83 . 324235 788336958.52 2 - . 2 62 4.45.64824744 +5244.43. - 3 4 4.48.39493243 2457.45..3 . - . 2 3.9834.2+.32+47453 53 .2256 - .3423.2.3953+5+53224 75 2 2. .. 0.35+ . 22.. 4+539+66 734 2.. 2..2. .2. - 2 .3 8735353 6624852. 63 . . - .327+35544562 2..2.2 . 2 . - 37233454.3 .2 .42 2 - 4267 .434.. 0.00+ +---------+---------+---------+---------+---------+post_p1 0.00 0.20 0.40 0.60 0.80 1.00 TYPE 'y' AND RETURN TO SEE PLOTS OF THE MARGINAL DISTRIBUTIONS OF P1 AND P2: y Posterior distribution of proportion P1: Each dot represents 3 points . . : : :: : .. . : . :::::::: .: .:. :.. :.: : :::..::::::::.::.::::.::: ::: : : ::::::::::::::::::::::::::::::::::: . :: :::::::::::::::::::::::::::::::::::: :.: :.::::::::::::::::::::::::::::::::::::::::::::. +---------+---------+---------+---------+---------+-------post_p1 0.00 0.20 0.40 0.60 0.80 1.00 Variable N Mean Median TrMean StDev SEMean post_p1 1000 0.49238 0.47582 0.49174 0.22129 0.00700 Variable Min Max Q1 Q3 post_p1 0.03479 0.95686 0.32448 0.67783 Posterior distribution of proportion P2: Each dot represents 3 points . : :. : . . . .: :::: :. : . : : :. ::. :::: ::: : . : : :.:.:: ::: ::::. :::. : .:: .:. ::.:.::::::::::.:::::::::::::::::..... ::: :::::::::::::::::::::::::::::::::::::: .::::::::::::::::::::::::::::::::::::::::::::::. +---------+---------+---------+---------+---------+-------post_p2 0.00 0.20 0.40 0.60 0.80 1.00 Variable N Mean Median TrMean StDev SEMean post_p2 1000 0.50041 0.50506 0.50049 0.22162 0.00701 Variable Min Max Q1 Q3 post_p2 0.04348 0.97014 0.32681 0.68286 TYPE 'y' AND RETURN TO SEE PLOT OF THE DISTRIBUTION OF P2-P1: y Each dot represents 5 points . :.: : : :::: : :.:::::::: . . :::::::::: .. :.: :::::::::::..:: .....:::::::::::::::::::: . . .. ....:::::::::::::::::::::::::::::.:::.... . +---------+---------+---------+---------+---------+-------p2-p1 -0.60 -0.40 -0.20 0.00 0.20 0.40 Variable N Mean Median TrMean StDev SEMean p2-p1 1000 0.00802 0.01194 0.00780 0.14468 0.00458 Variable Min Max Q1 Q3 p2-p1 -0.46672 0.43172 -0.07884 0.09405 MTB > exec 'pp_exch' MTB > ################################################################## MTB > # MACRO 'PP_EXCH' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT 2 BINOMIAL PROPORTIONS # MTB > # USING AN EXCHANGEABLE PRIOR FOR P1 AND P2. # MTB > ################################################################## T INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN 1ST GROUP: DATA> 4 16 A INPUT OBSERVED NUMBER OF SUCCESSES AND FAILURES IN 2ND GROUP: DATA> 8 12 INPUT STANDARD DEVIATION OF THE LOGITS T1 AND T2: DATA> .5 INPUT NUMBER OF SIMULATED VALUES: DATA> 1000 TYPE 'y' AND RETURN TO SEE A PLOT OF THE JOINT DISTRIBUTION OF P1 AND P2: y 1.05+ - post_p2 - - - 0.70+ . . .. - 3 2 5 - 4 36 465.. 2 - 7 79+427+34.4253 3 - 4+9+ +++++8+266.2.. 0.35+ 2 45+ ++++9+++8844542.. - 76+++4++++++3+52..3 - 2487367.+6. 42 . - . 22. 2. - 0.00+ +---------+---------+---------+---------+---------+post_p1 0.00 0.20 0.40 0.60 0.80 1.00 TYPE 'y' AND RETURN TO SEE PLOTS OF THE MARGINAL DISTRIBUTIONS OF P1 AND P2: y Posterior distribution of proportion P1: Each dot represents 5 points . : . : . : : : : : . ::::: :: .. : :::::::.:: . . :: :::::::::::::::. : .: .:::::::::::::::::::.:.. .. .::::::::::::::::::::::::::::::::..:::.... .. -------+---------+---------+---------+---------+---------post_p1 0.10 0.20 0.30 0.40 0.50 0.60 Variable N Mean Median TrMean StDev SEMean post_p1 1000 0.27170 0.26398 0.26881 0.07889 0.00249 Variable Min Max Q1 Q3 post_p1 0.07808 0.54887 0.21711 0.31722 Posterior distribution of proportion P2: Each dot represents 7 points . : : : . ...:.: ::..: :.:::::: ::::: . . .:::::::::.:::::::.. : .. ...:::::::::::::::::::::::::::::.... . .. ... . -+---------+---------+---------+---------+---------+-----post_p2 0.12 0.24 0.36 0.48 0.60 0.72 Variable N Mean Median TrMean StDev SEMean post_p2 1000 0.35826 0.35684 0.35544 0.08964 0.00283 Variable Min Max Q1 Q3 post_p2 0.12286 0.72539 0.29329 0.41049 TYPE 'y' AND RETURN TO SEE PLOT OF THE DISTRIBUTION OF P2-P1: y Each dot represents 5 points : . : :: : : : : .::..: : : : : . ::::::. : : :::.:::::::::::.: : : .::.::::::::::::::::::::. .: . .........::::::::::::::::::::::::::::::.:.. : . . . . -------+---------+---------+---------+---------+---------p2-p1 -0.12 0.00 0.12 0.24 0.36 0.48 Variable N Mean Median TrMean StDev SEMean p2-p1 1000 0.08656 0.08873 0.08615 0.10264 0.00325 Variable Min Max Q1 Q3 p2-p1 -0.21141 0.44013 0.01937 0.15436 **************************** CHAPTER 6 ********************************* MTB > name c1 'm' c2 'prior' MTB > set 'm' DATA> 174 176 178 DATA> end MTB > set 'prior' DATA> .333 .333 .333 DATA> name c3 'data' MTB > set 'data' DATA> 182 172 173 176 176 180 173 174 179 175 DATA> end MTB > MTB > exec 'm_disc' MTB > ################################################################## MTB > # MACRO 'M_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A NORMAL MEAN M # MTB > # USING A DISCRETE SET OF M MODELS. # MTB > # -------------------------------------------------------------- # MTB > # INPUT: VALUES OF M IN COLUMN 'M' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT POPULATION STANDARD DEVIATION: DATA> 3 OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN AND SAMPLE SIZE: y DATA> 3 OBS_DATA 182 172 173 176 176 180 173 174 179 175 Mean of data = 176.00 Total number of observations in data = 10 Row m prior M_x_PRIO LIKE PRODUCT POST M_x_POST 1 174 0.333333 58.000 108368 36123 0.089065 15.497 2 176 0.333333 58.667 1000000 333333 0.821871 144.649 3 178 0.333333 59.333 108368 36123 0.089065 15.853 4 176.000 176.000 - 0.90+ - * PROB - * - 2 - * 0.60+ 2 - * - 2 - * - * 0.30+ 2 - * - 2 - * - + 2 + 0.00+ 7 * 7 --------+---------+---------+---------+---------+--------MODEL 174.40 175.20 176.00 176.80 177.60 MTB > exec 'normal_s' MTB > ################################################################## MTB > # MACRO 'NORMAL_S' # MTB > # -------------------------------------------------------------- # MTB > # FINDS THE NORMAL(A,B) PRIOR DISTRIBUTION # MTB > # WHICH MATCHES TWO QUANTILES. # MTB > ################################################################## Input the first probability P1 and quantile M1: DATA> .5 174 Input the second probability P2 and quantile M2: DATA> .9 180 The matching values of the normal density parameters corresponding to your 2 quantiles are given by: MEAN 174 STD 4.68183 MTB > exec 'm_cont' MTB > ################################################################## MTB > # MACRO 'M_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A NORMAL MEAN M # MTB > # USING A CONTINUOUS PRIOR FOR M. # MTB > ################################################################## DO YOU WISH TO USE A FLAT PRIOR DENSITY FOR M? (TYPE 'y' OR 'n'.) IF NO, INPUT MEAN AND STANDARD DEVIATION FOR THE PRIOR DENSITY.. n DATA> 174 4.68 PR_MEAN 174 PR_STD 4.68 OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN, STANDARD DEVIATION, AND SAMPLE SIZE. y DATA> 3 OBS_DATA 182 172 173 176 176 180 173 174 179 175 MEAN 176 STD 3.33333 COUNT 10 THE POSTERIOR DENSITY FOR M IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 175.877 1.162 THE PREDICTIVE DENSITY OF THE NEXT OBSERVATION IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 175.877 3.969 MTB > exec 'normal' MTB > ################################################################## MTB > # MACRO 'NORMAL' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # FOR A NORMAL(M,S) DISTRIBUTION, GRAPHS DENSITY CURVE, # MTB > # AND COMPUTES CUMULATIVE PROBABILITIES AND QUANTILES. # MTB > ################################################################## INPUT THE VALUES OF THE MEAN AND STANDARD DEVIATION OF THE NORMAL DISTRIBUTION: DATA> 175.877 1.162 TYPE 'y' TO SEE A PLOT OF THE NORMAL DENSITY: y - 0.36+ - ****** density - * * - * * - * * 0.24+ * * - * * - * * - * * - * * 0.12+ * * - ** ** - * * - ** ** - **** **** 0.00+ **** **** ------+---------+---------+---------+---------+---------+M 172.5 174.0 175.5 177.0 178.5 180.0 TYPE 'y' TO COMPUTE CUMULATIVE PROBABILITIES: ---------------------------------------------------------------- Input values of M of interest. The output is the column of values M and the column of cumulative probabilities PROB_LT. ---------------------------------------------------------------- y DATA> 170 180 190 DATA> end Row M PROB_LT 1 170 0.00000 2 180 0.99981 3 190 1.00000 TYPE 'y' TO COMPUTE QUANTILES: ---------------------------------------------------------------- Input probabilities for which you wish to compute quantiles. The output is the probabilities in the column PROB and the corresponding quantiles in the column QUANTILE. ---------------------------------------------------------------- y DATA> .025 .975 DATA> end Row PROB QUANTILE 1 0.025 173.600 2 0.975 178.154 MTB > exec 'm_norm_t' MTB > ################################################################## MTB > # MACRO 'M_NORM_T' # MTB > # -------------------------------------------------------------- # MTB > # TESTS THE HYPOTHESIS THAT M = M0 USING A NORMAL PRIOR # MTB > ################################################################## ENTER THE NULL HYPOTHESIS MEAN M0: DATA> 170 ENTER THE PRIOR PROBABILITY OF M0: DATA> .5 FOR THE ALTERNATIVE HYPOTHESIS THAT M = M0, ENTER STANDARD DEVIATION(S) OF THE NORMAL PRIOR DISTRIBUTION: DATA> .5 1 2 4 8 DATA> end ENTER THE STANDARD DEVIATION OF THE POPULATION: DATA> 3 OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN AND SAMPLE SIZE: y DATA> 3 OBS_DATA 182 172 173 176 176 180 173 174 179 175 Mean of OBS_DATA= 176.00 Total number of observations in OBS_DATA= 10 The Bayes factor in favor of the null hypothesis is: BF_HK 0.000000 0.000092 0.018964 0.138641 0.155533 The Bayes factor against the null hypothesis is: BF_KH 4070108 10848 53 7 6 The posterior probability of the null hypothesis is: prob_H 0.000000 0.000092 0.018611 0.121760 0.134598 MTB > exec 'm_nchi' MTB > ################################################################## MTB > # MACRO 'M_NCHI' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A NORMAL MEAN M AND STANDARD DEVIATION S # MTB > # EXACT NORMAL/CHI-SQUARED INFERENCE # MTB > ################################################################## OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN, STANDARD DEVIATION, AND SAMPLE SIZE. y DATA> 3 OBS_DATA 182 172 173 176 176 180 173 174 179 175 MEAN 176 STD 3.33333 COUNT 10 The mean M has a t(m,se,df) distribution with Row m se df 1 176 1.05409 9 A 95% probability interval for M is: m_int 173.615 178.385 The standard deviation S has a inverse chi-square(S,df) distribution with Row s df 1 100 9 A 95% probability interval for S is: s_int 2.29278 6.08537 Type 'y' and return to see plots of marginal posterior densities for M and S: y - 1.05+ - ***** density - * * - - * * 0.70+ * * - * * - - * * - * * 0.35+ * * - * * - ** ** - ** ** - **** **** 0.00+ ******* ******* --------+---------+---------+---------+---------+--------m 172.8 174.4 176.0 177.6 179.2 - 1.05+ - *** density - * * - * * - * 0.70+ * - * - * - * * - * 0.35+ ** - * * - ** - * *** - * ****** 0.00+ ***** *************** ------+---------+---------+---------+---------+---------+s 1.5 3.0 4.5 6.0 7.5 9.0 **************************** CHAPTER 7 ********************************* MTB > name c1 'smoke' c2 'quit' MTB > set 'smoke' DATA> 4.5 5.4 5.6 5.9 6 6.1 6.4 6.6 6.6 6.6 6.9 6.9 DATA> 7.1 7.1 7.2 7.5 7.6 7.6 7.8 8 9.9 DATA> end MTB > set 'quit' DATA> 5.4 6.6 6.8 6.8 6.9 7.2 7.3 7.4 DATA> end MTB > exec 'm_cont' MTB > ################################################################## MTB > # MACRO 'M_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A NORMAL MEAN M # MTB > # USING A CONTINUOUS PRIOR FOR M. # MTB > ################################################################## DO YOU WISH TO USE A FLAT PRIOR DENSITY FOR M? (TYPE 'y' OR 'n'.) IF NO, INPUT MEAN AND STANDARD DEVIATION FOR THE PRIOR DENSITY.. n DATA> 7.7 .7 PR_MEAN 7.7 PR_STD 0.7 OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN, STANDARD DEVIATION, AND SAMPLE SIZE. y DATA> 1 OBS_DATA 4.5 5.4 5.6 5.9 6.0 6.1 6.4 6.6 6.6 6.6 6.9 6.9 7.1 7.1 7.2 7.5 7.6 7.6 7.8 8.0 9.9 MEAN 6.82381 STD 1.11978 COUNT 21 THE POSTERIOR DENSITY FOR M IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 6.92242 0.23484 THE PREDICTIVE DENSITY OF THE NEXT OBSERVATION IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 6.92242 1.16624 MTB > exec 'm_cont' MTB > ################################################################## MTB > # MACRO 'M_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT A NORMAL MEAN M # MTB > # USING A CONTINUOUS PRIOR FOR M. # MTB > ################################################################## DO YOU WISH TO USE A FLAT PRIOR DENSITY FOR M? (TYPE 'y' OR 'n'.) IF NO, INPUT MEAN AND STANDARD DEVIATION FOR THE PRIOR DENSITY.. n DATA> 7.7 .7 PR_MEAN 7.7 PR_STD 0.7 OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) IF YES, INPUT NUMBER OF COLUMN. IF NO, INPUT OBSERVED SAMPLE MEAN, STANDARD DEVIATION, AND SAMPLE SIZE. y DATA> 2 OBS_DATA 5.4 6.6 6.8 6.8 6.9 7.2 7.3 7.4 MEAN 6.8 STD 0.630193 COUNT 8 THE POSTERIOR DENSITY FOR M IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 6.91923 0.25478 THE PREDICTIVE DENSITY OF THE NEXT OBSERVATION IS NORMAL WITH MEAN AND STANDARD DEVIATION: MEAN STD 6.91923 0.81458 MTB > exec 'mm_cont' MTB > ################################################################## MTB > # MACRO 'MM_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT THE DIFFERENCE IN NORMAL MEANS M1 AND M2 # MTB > ################################################################## INPUT MEAN AND STANDARD DEVIATION FOR NORMAL DISTRIBUTION FOR MEAN M1: DATA> 6.92242 0.23484 INPUT MEAN AND STANDARD DEVIATION FOR NORMAL DISTRIBUTION FOR MEAN M2: DATA> 6.91923 0.25478 6.91923 0.25478 THE POSTERIOR DENSITY FOR M1-M2 IS NORMAL WITH MEAN AND STANDARD DEVIATION: Row mn st 1 0.0031900 0.346501 MTB > exec 'mm_tt' MTB > ################################################################## MTB > # MACRO 'MM_TT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE ABOUT THE DIFFERENCE IN TWO NORMAL MEANS # MTB > # USING A CONTINUOUS PRIOR FOR M. # MTB > ################################################################## OBSERVED DATA IN WORKSHEET? (TYPE 'y' OR 'n'.) y NOTE: INPUT NUMBER OF COLUMN OF FIRST DATASET DATA> 1 NOTE: INPUT NUMBER OF COLUMN OF SECOND DATASET DATA> 2 INPUT NUMBER OF SIMULATED VALUES: DATA> 500 Simulated values of M1 and M2: Each dot represents 2 points . . :::. : :::: :. ::::..: :.::.:::::::: ::::::::::::: .::::::::::::: . . :::::::::::::::::: : ...:::::::::::::::::: : : . ....::::::::::::::::::::::::::.::.: . . -----+---------+---------+---------+---------+---------+-m1 Each dot represents 3 points . : . .::.: ::.:::::. . .:::::::::..: .:::::::::::::: .:::::::::::::::.:. .. . .. :.::::::::::::::::::::::::::....... . . -----+---------+---------+---------+---------+---------+-m2 6.00 6.40 6.80 7.20 7.60 8.00 Simulated values of M2-M1: Each dot represents 2 points : . : . : :.: ::.: : :::::::: .: : ::::::::.::: ::::::::::::::: . . ::::::::::::::::: : . .:.:::::::::::::::::::::::. .....: ::::::::::::::::::::::::::::::.:.... .. -----+---------+---------+---------+---------+---------+-m_diff -1.00 -0.50 0.00 0.50 1.00 1.50 Variable N Mean Median TrMean StDev SEMean m_diff 500 -0.0124 -0.0141 -0.0139 0.3640 0.0163 Variable Min Max Q1 Q3 m_diff -1.0973 1.2697 -0.2376 0.2257 **************************** CHAPTER 8 ********************************* MTB > name c1 'temp' c2 'freq' MTB > set 'temp' DATA> 89 72 93 84 81 75 70 82 69 83 DATA> end MTB > set 'freq' DATA> 20 16 20 18 17 16 15 17 15 16 DATA> end MTB > exec 'lin_reg' INPUT COLUMN NUMBERS OF X AND Y DATA: DATA> 2 1 THE LEAST-SQUARES LINE HAS SLOPE AND INTERCEPT: Row B A 1 4.06665 10.6669 THE POSTERIOR DENSITY FOR b IS NORMAL WITH MEAN AND STANDARD DEVIATION: Row MEAN STD 1 4.06665 0.669801 Input 'y' and 'return' to obtain prediction intervals: y NOTE: FOR PREDICTING Y FOR GIVEN VALUES OF X, INPUT X VALUES OF INTEREST: DATA> 16 18 20 DATA> end NOTE: THE PREDICTIVE DENSITY OF THE NEXT OBSERVATION FOR DIFFERENT VALUES OF X IS NORMAL WITH MEANS AND STANDARD DEVIATIONS GIVEN BELOW: Row X_ MEAN_Y STD_Y 1 16 75.7334 3.90558 2 18 83.8667 3.90558 3 20 92.0000 4.34081 MTB > read c1-c3 DATA> 11 68 3 DATA> 9 23 5 DATA> end MTB > exec 'c_table' MTB > ################################################################## MTB > # MACRO 'C_TABLE' # MTB > # -------------------------------------------------------------- # MTB > # BAYES TEST OF INDEPENDENCE FOR A 2-WAY CONTINGENCY TABLE # MTB > # -------------------------------------------------------------- # MTB > # INPUT: TABLE IN CONSECUTIVE COLUMNS OF WORKSHEET # MTB > # OUTPUT: BAYES FACTOR AGAINST THE HYPOTHESIS OF INDEPENDENCE # MTB > ################################################################## INPUT THE NUMBER OF THE FIRST COLUMN WHICH CONTAINS THE CONTINGENCY TABLE: DATA> 1 INPUT THE NUMBER OF COLUMNS OF THE TABLE: DATA> 3 Expected counts are printed below observed counts C1 C2 C3 Total 1 11 68 3 82 13.78 62.71 5.51 2 9 23 5 37 6.22 28.29 2.49 Total 20 91 8 119 ChiSq = 0.561 + 0.447 + 1.145 + 1.244 + 0.991 + 2.538 = 6.926 df = 2, p = 0.032 1 cells with expected counts less than 5.0 ------------------------------------------------------- The Bayes factor against the hypothesis of independence with uniform priors is: BAYES_F 1.66221 ------------------------------------------------------- **************************** CHAPTER 9 ********************************* MTB > name c1 'model' c2 'prior' MTB > set 'model' DATA> .5 .25 .125 DATA> end MTB > set 'prior' DATA> .2 .5 .3 DATA> end MTB > exec 'mod_disc' MTB > ################################################################## MTB > # MACRO 'MOD_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING A FINITE COLLECTION OF MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF MODEL IN 'MODEL' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 3 INPUT (sample sum, time interval) DATA> 10 24 Row model prior LIKE PRODUCT POST 1 0.500 0.2 1000000 200000 0.500870 2 0.250 0.5 393974 196987 0.493324 3 0.125 0.3 7728 2318 0.005806 PRIOR MEAN OF MODELS: MEAN 0.2625 POSTERIOR MEAN OF MODELS: MEAN 0.374492 Storing in file: MINITAB.MTB * NOTE * Existing file replaced. Executing from file: MINITAB.MTB - * 0.48+ 2 * - * * PROB - * 2 - 2 * - * * 0.32+ * * - * 2 - 2 * - * * - * * 0.16+ 2 2 - * * - * * - 2 2 - * * 0.00+ + * * ----+---------+---------+---------+---------+---------+--MODELS 0.140 0.210 0.280 0.350 0.420 0.490 MTB > name c1 'model' c2 'prior' MTB > set 'model' DATA> 1:200 DATA> end MTB > let 'prior'=.005+0*'model' MTB > exec 'mod_disc' MTB > ################################################################## MTB > # MACRO 'MOD_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING A FINITE COLLECTION OF MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF MODEL IN 'MODEL' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 5 INPUT (maximum observation, sample size) DATA> 100 5 Executing from file: lk_unf_n.MTB Row model prior LIKE PRODUCT POST 1 1 0.005 0 0.00 0.0000000 2 2 0.005 0 0.00 0.0000000 3 3 0.005 0 0.00 0.0000000 4 4 0.005 0 0.00 0.0000000 5 5 0.005 0 0.00 0.0000000 6 6 0.005 0 0.00 0.0000000 7 7 0.005 0 0.00 0.0000000 8 8 0.005 0 0.00 0.0000000 9 9 0.005 0 0.00 0.0000000 10 10 0.005 0 0.00 0.0000000 11 11 0.005 0 0.00 0.0000000 12 12 0.005 0 0.00 0.0000000 13 13 0.005 0 0.00 0.0000000 14 14 0.005 0 0.00 0.0000000 15 15 0.005 0 0.00 0.0000000 16 16 0.005 0 0.00 0.0000000 17 17 0.005 0 0.00 0.0000000 18 18 0.005 0 0.00 0.0000000 19 19 0.005 0 0.00 0.0000000 20 20 0.005 0 0.00 0.0000000 21 21 0.005 0 0.00 0.0000000 22 22 0.005 0 0.00 0.0000000 23 23 0.005 0 0.00 0.0000000 24 24 0.005 0 0.00 0.0000000 25 25 0.005 0 0.00 0.0000000 26 26 0.005 0 0.00 0.0000000 27 27 0.005 0 0.00 0.0000000 28 28 0.005 0 0.00 0.0000000 29 29 0.005 0 0.00 0.0000000 30 30 0.005 0 0.00 0.0000000 31 31 0.005 0 0.00 0.0000000 32 32 0.005 0 0.00 0.0000000 33 33 0.005 0 0.00 0.0000000 34 34 0.005 0 0.00 0.0000000 35 35 0.005 0 0.00 0.0000000 36 36 0.005 0 0.00 0.0000000 37 37 0.005 0 0.00 0.0000000 38 38 0.005 0 0.00 0.0000000 39 39 0.005 0 0.00 0.0000000 40 40 0.005 0 0.00 0.0000000 41 41 0.005 0 0.00 0.0000000 42 42 0.005 0 0.00 0.0000000 43 43 0.005 0 0.00 0.0000000 44 44 0.005 0 0.00 0.0000000 45 45 0.005 0 0.00 0.0000000 46 46 0.005 0 0.00 0.0000000 47 47 0.005 0 0.00 0.0000000 48 48 0.005 0 0.00 0.0000000 49 49 0.005 0 0.00 0.0000000 50 50 0.005 0 0.00 0.0000000 51 51 0.005 0 0.00 0.0000000 52 52 0.005 0 0.00 0.0000000 53 53 0.005 0 0.00 0.0000000 54 54 0.005 0 0.00 0.0000000 55 55 0.005 0 0.00 0.0000000 56 56 0.005 0 0.00 0.0000000 57 57 0.005 0 0.00 0.0000000 58 58 0.005 0 0.00 0.0000000 59 59 0.005 0 0.00 0.0000000 60 60 0.005 0 0.00 0.0000000 61 61 0.005 0 0.00 0.0000000 62 62 0.005 0 0.00 0.0000000 63 63 0.005 0 0.00 0.0000000 64 64 0.005 0 0.00 0.0000000 65 65 0.005 0 0.00 0.0000000 66 66 0.005 0 0.00 0.0000000 67 67 0.005 0 0.00 0.0000000 68 68 0.005 0 0.00 0.0000000 69 69 0.005 0 0.00 0.0000000 70 70 0.005 0 0.00 0.0000000 71 71 0.005 0 0.00 0.0000000 72 72 0.005 0 0.00 0.0000000 73 73 0.005 0 0.00 0.0000000 74 74 0.005 0 0.00 0.0000000 75 75 0.005 0 0.00 0.0000000 76 76 0.005 0 0.00 0.0000000 77 77 0.005 0 0.00 0.0000000 78 78 0.005 0 0.00 0.0000000 79 79 0.005 0 0.00 0.0000000 80 80 0.005 0 0.00 0.0000000 81 81 0.005 0 0.00 0.0000000 82 82 0.005 0 0.00 0.0000000 83 83 0.005 0 0.00 0.0000000 84 84 0.005 0 0.00 0.0000000 85 85 0.005 0 0.00 0.0000000 86 86 0.005 0 0.00 0.0000000 87 87 0.005 0 0.00 0.0000000 88 88 0.005 0 0.00 0.0000000 89 89 0.005 0 0.00 0.0000000 90 90 0.005 0 0.00 0.0000000 91 91 0.005 0 0.00 0.0000000 92 92 0.005 0 0.00 0.0000000 93 93 0.005 0 0.00 0.0000000 94 94 0.005 0 0.00 0.0000000 95 95 0.005 0 0.00 0.0000000 96 96 0.005 0 0.00 0.0000000 97 97 0.005 0 0.00 0.0000000 98 98 0.005 0 0.00 0.0000000 99 99 0.005 0 0.00 0.0000000 100 100 0.005 1000000 5000.00 0.0417410 101 101 0.005 951468 4757.34 0.0397152 102 102 0.005 905732 4528.66 0.0378061 103 103 0.005 862610 4313.05 0.0360062 104 104 0.005 821928 4109.64 0.0343081 105 105 0.005 783527 3917.64 0.0327052 106 106 0.005 747259 3736.30 0.0311913 107 107 0.005 712988 3564.94 0.0297608 108 108 0.005 680584 3402.92 0.0284083 109 109 0.005 649931 3249.66 0.0271288 110 110 0.005 620921 3104.61 0.0259179 111 111 0.005 593452 2967.26 0.0247713 112 112 0.005 567428 2837.14 0.0236850 113 113 0.005 542760 2713.80 0.0226554 114 114 0.005 519369 2596.85 0.0216790 115 115 0.005 497177 2485.89 0.0207527 116 116 0.005 476114 2380.57 0.0198735 117 117 0.005 456111 2280.56 0.0190385 118 118 0.005 437109 2185.55 0.0182454 119 119 0.005 419050 2095.25 0.0174916 120 120 0.005 401878 2009.39 0.0167748 121 121 0.005 385543 1927.72 0.0160930 122 122 0.005 370000 1850.00 0.0154442 123 123 0.005 355202 1776.01 0.0148265 124 124 0.005 341108 1705.54 0.0142382 125 125 0.005 327680 1638.40 0.0136777 126 126 0.005 314882 1574.41 0.0131435 127 127 0.005 302679 1513.39 0.0126341 128 128 0.005 291039 1455.19 0.0121483 129 129 0.005 279932 1399.66 0.0116846 130 130 0.005 269329 1346.65 0.0112421 131 131 0.005 259205 1296.03 0.0108195 132 132 0.005 249535 1247.67 0.0104158 133 133 0.005 240294 1201.47 0.0100301 134 134 0.005 231460 1157.30 0.0096614 135 135 0.005 223014 1115.07 0.0093088 136 136 0.005 214934 1074.67 0.0089716 137 137 0.005 207204 1036.02 0.0086489 138 138 0.005 199804 999.02 0.0083400 139 139 0.005 192720 963.60 0.0080443 140 140 0.005 185935 929.67 0.0077611 141 141 0.005 179434 897.17 0.0074897 142 142 0.005 173204 866.02 0.0072297 143 143 0.005 167232 836.16 0.0069804 144 144 0.005 161506 807.53 0.0067414 145 145 0.005 156013 780.06 0.0065121 146 146 0.005 150743 753.71 0.0062921 147 147 0.005 145685 728.42 0.0060810 148 148 0.005 140829 704.15 0.0058783 149 149 0.005 136166 680.83 0.0056837 150 150 0.005 131687 658.44 0.0054968 151 151 0.005 127384 636.92 0.0053171 152 152 0.005 123249 616.24 0.0051445 153 153 0.005 119273 596.37 0.0049786 154 154 0.005 115451 577.25 0.0048190 155 155 0.005 111774 558.87 0.0046656 156 156 0.005 108237 541.19 0.0045179 157 157 0.005 104834 524.17 0.0043759 158 158 0.005 101558 507.79 0.0042391 159 159 0.005 98404 492.02 0.0041075 160 160 0.005 95367 476.84 0.0039807 161 161 0.005 92443 462.21 0.0038586 162 162 0.005 89624 448.12 0.0037410 163 163 0.005 86909 434.54 0.0036277 164 164 0.005 84291 421.46 0.0035184 165 165 0.005 81767 408.84 0.0034131 166 166 0.005 79334 396.67 0.0033115 167 167 0.005 76987 384.94 0.0032135 168 168 0.005 74723 373.62 0.0031190 169 169 0.005 72538 362.69 0.0030278 170 170 0.005 70430 352.15 0.0029398 171 171 0.005 68394 341.97 0.0028549 172 172 0.005 66429 332.15 0.0027728 173 173 0.005 64531 322.66 0.0026936 174 174 0.005 62698 313.49 0.0026171 175 175 0.005 60927 304.64 0.0025432 176 176 0.005 59216 296.08 0.0024717 177 177 0.005 57562 287.81 0.0024027 178 178 0.005 55963 279.81 0.0023359 179 179 0.005 54417 272.09 0.0022714 180 180 0.005 52922 264.61 0.0022090 181 181 0.005 51476 257.38 0.0021487 182 182 0.005 50078 250.39 0.0020903 183 183 0.005 48724 243.62 0.0020338 184 184 0.005 47415 237.07 0.0019791 185 185 0.005 46147 230.73 0.0019262 186 186 0.005 44920 224.60 0.0018750 187 187 0.005 43731 218.66 0.0018254 188 188 0.005 42581 212.90 0.0017774 189 189 0.005 41466 207.33 0.0017308 190 190 0.005 40386 201.93 0.0016858 191 191 0.005 39340 196.70 0.0016421 192 192 0.005 38326 191.63 0.0015998 193 193 0.005 37343 186.72 0.0015588 194 194 0.005 36391 181.95 0.0015190 195 195 0.005 35467 177.34 0.0014804 196 196 0.005 34572 172.86 0.0014431 197 197 0.005 33703 168.52 0.0014068 198 198 0.005 32861 164.30 0.0013716 199 199 0.005 32043 160.22 0.0013375 200 200 0.005 31250 156.25 0.0013044 PRIOR MEAN OF MODELS: MEAN 100.5 POSTERIOR MEAN OF MODELS: MEAN 123.976 - 0.045+ - * PROB - 4* - 22 - 37 0.030+ 373 - 379* - 3787 - 37896 - 368++5 0.015+ 379+++8 - 2789++++3 - 478+++++++6 - 278+++++++++++4 - 4888+++++++++++++++++++8* 0.000+ ++++++++++++++++++++++++++47888++++++++++++++++++++ +---------+---------+---------+---------+---------+------MODELS 0 40 80 120 160 200 MTB > exec 'disc_sum' INPUT NUMBER OF COLUMN WHICH CONTAINS VALUES OF VARIABLE: DATA> 1 INPUT NUMBER OF COLUMN WHICH CONTAINS PROBABILITIES: DATA> 51 TYPE 'y' TO SEE A PLOT OF THE PROBABILITIES: n TYPE 'y' TO GET SUMMARIES OF THE DISTRIBUTION: y Row MODE MEAN STD 1 100 123.976 22.7748 TYPE 'y' TO COMPUTE CUMULATIVE PROBABILITIES: -------------------------------------------------------------------- Input values of variable of interest. The output is the column of values and the column of cumulative probabilities PROB_LE. -------------------------------------------------------------------- y DATA> 150 DATA> end Row VALUE PROB_LE 1 150 0.861175 TYPE 'y' TO COMPUTE PROBABILITY INTERVALS: -------------------------------------------------------------------- Input list of probabilities. For each probability p, the set of values of the variable for which the probability content of the set exceeds p is given. -------------------------------------------------------------------- y DATA> .9 DATA> end PROB_SET 0.90334 SET 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 -------------------------------------------------------------------------- MTB > name c1 'model' c2 'prior' MTB > set 'model' DATA> 0:100 DATA> end MTB > let 'prior'=0*'model'+1/101 MTB > exec 'mod_disc' MTB > ################################################################## MTB > # MACRO 'MOD_DISC' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING A FINITE COLLECTION OF MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF MODEL IN 'MODEL' AND PRIOR PROBABILITIES # MTB > # IN COLUMN 'PRIOR' # MTB > # OUTPUT: POSTERIOR PROBABILITIES IN COLUMN 'POST' # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 4 INPUT (population size, sample size, number of successes) DATA> 100 20 12 Row model prior LIKE PRODUCT POST 1 0 0.009901 0 0.00 0.0000000 2 1 0.009901 0 0.00 0.0000000 3 2 0.009901 0 0.00 0.0000000 4 3 0.009901 0 0.00 0.0000000 5 4 0.009901 0 0.00 0.0000000 6 5 0.009901 0 0.00 0.0000000 7 6 0.009901 0 0.00 0.0000000 8 7 0.009901 0 0.00 0.0000000 9 8 0.009901 0 0.00 0.0000000 10 9 0.009901 0 0.00 0.0000000 11 10 0.009901 0 0.00 0.0000000 12 11 0.009901 0 0.00 0.0000000 13 12 0.009901 0 0.00 0.0000000 14 13 0.009901 0 0.00 0.0000000 15 14 0.009901 0 0.00 0.0000000 16 15 0.009901 0 0.00 0.0000000 17 16 0.009901 1 0.01 0.0000000 18 17 0.009901 2 0.02 0.0000001 19 18 0.009901 6 0.06 0.0000003 20 19 0.009901 15 0.15 0.0000006 21 20 0.009901 34 0.34 0.0000014 22 21 0.009901 72 0.71 0.0000030 23 22 0.009901 141 1.40 0.0000059 24 23 0.009901 265 2.63 0.0000111 25 24 0.009901 475 4.70 0.0000198 26 25 0.009901 817 8.09 0.0000341 27 26 0.009901 1355 13.42 0.0000566 28 27 0.009901 2175 21.54 0.0000908 29 28 0.009901 3389 33.55 0.0001414 30 29 0.009901 5137 50.86 0.0002144 31 30 0.009901 7596 75.21 0.0003171 32 31 0.009901 10976 108.67 0.0004581 33 32 0.009901 15522 153.69 0.0006479 34 33 0.009901 21521 213.08 0.0008983 35 34 0.009901 29286 289.96 0.0012224 36 35 0.009901 39159 387.71 0.0016345 37 36 0.009901 51505 509.95 0.0021498 38 37 0.009901 66693 660.33 0.0027838 39 38 0.009901 85090 842.47 0.0035517 40 39 0.009901 107043 1059.83 0.0044680 41 40 0.009901 132862 1315.46 0.0055457 42 41 0.009901 162783 1611.71 0.0067946 43 42 0.009901 196990 1950.39 0.0082225 44 43 0.009901 235547 2332.15 0.0098318 45 44 0.009901 278407 2756.51 0.0116209 46 45 0.009901 325411 3221.89 0.0135828 47 46 0.009901 376213 3724.88 0.0157033 48 47 0.009901 430338 4260.77 0.0179625 49 48 0.009901 487160 4823.37 0.0203343 50 49 0.009901 545885 5404.80 0.0227855 51 50 0.009901 605615 5996.19 0.0252787 52 51 0.009901 665240 6586.53 0.0277675 53 52 0.009901 723622 7164.57 0.0302044 54 53 0.009901 779502 7717.85 0.0325368 55 54 0.009901 831626 8233.93 0.0347125 56 55 0.009901 878708 8700.08 0.0366777 57 56 0.009901 919530 9104.26 0.0383817 58 57 0.009901 952999 9435.64 0.0397787 59 58 0.009901 978043 9683.60 0.0408241 60 59 0.009901 993923 9840.82 0.0414869 61 60 0.009901 1000000 9900.99 0.0417405 62 61 0.009901 995919 9860.59 0.0415702 63 62 0.009901 981639 9719.20 0.0409741 64 63 0.009901 957364 9478.86 0.0399609 65 64 0.009901 923559 9144.15 0.0385498 66 65 0.009901 880990 8722.68 0.0367730 67 66 0.009901 830701 8224.76 0.0346739 68 67 0.009901 773867 7662.05 0.0323016 69 68 0.009901 711925 7048.76 0.0297161 70 69 0.009901 646388 6399.88 0.0269806 71 70 0.009901 578831 5731.00 0.0241607 72 71 0.009901 510847 5057.90 0.0213231 73 72 0.009901 443947 4395.52 0.0185306 74 73 0.009901 379543 3757.85 0.0158423 75 74 0.009901 318811 3156.54 0.0133073 76 75 0.009901 262791 2601.89 0.0109691 77 76 0.009901 212234 2101.33 0.0088588 78 77 0.009901 167640 1659.81 0.0069974 79 78 0.009901 129240 1279.61 0.0053946 80 79 0.009901 96999 960.39 0.0040488 81 80 0.009901 70668 699.68 0.0029497 82 81 0.009901 49795 493.02 0.0020785 83 82 0.009901 33789 334.54 0.0014104 84 83 0.009901 21958 217.40 0.0009165 85 84 0.009901 13573 134.39 0.0005666 86 85 0.009901 7911 78.33 0.0003302 87 86 0.009901 4297 42.55 0.0001794 88 87 0.009901 2141 21.20 0.0000894 89 88 0.009901 957 9.48 0.0000400 90 89 0.009901 371 3.67 0.0000155 91 90 0.009901 118 1.17 0.0000049 92 91 0.009901 28 0.28 0.0000012 93 92 0.009901 1 0.01 0.0000001 94 93 0.009901 0 0.00 0.0000000 95 94 0.009901 0 0.00 0.0000000 96 95 0.009901 0 0.00 0.0000000 97 96 0.009901 0 0.00 0.0000000 98 97 0.009901 0 0.00 0.0000000 99 98 0.009901 0 0.00 0.0000000 100 99 0.009901 0 0.00 0.0000000 101 100 0.009901 0 0.00 0.0000000 PRIOR MEAN OF MODELS: MEAN 50 POSTERIOR MEAN OF MODELS: MEAN 59.2748 - 0.045+ - *22 PROB - *3323 - *433433 - 2232232* 0.030+ 244344343 - 4433223442 - 44432443344 - *443342233342 - 5553424334456 0.015+ 466442423334463 - 2874434222334669* - 2+866443444434458+ - 4++965443222244579++ - 9++++874444444444468+++6 0.000+ +++++++++++++++++++++975443222222234457++++++++++++ +---------+---------+---------+---------+---------+------MODELS 0 20 40 60 80 100 MTB > name c1 'model' c2 'prior1' c3 'prior2' MTB > set 'model' DATA> .2:.34/.02 DATA> end MTB > set 'prior1' DATA> .05 .05 .1 .25 .25 .15 .10 .05 DATA> set 'prior2' DATA> .2 .2 .2 .15 .1 .05 .05 .05 DATA> end MTB > prin c1-c3 Row model prior1 prior2 1 0.20 0.05 0.20 2 0.22 0.05 0.20 3 0.24 0.10 0.20 4 0.26 0.25 0.15 5 0.28 0.25 0.10 6 0.30 0.15 0.05 7 0.32 0.10 0.05 8 0.34 0.05 0.05 MTB > exec 'mod_crit' MTB > ################################################################## MTB > # MACRO 'MOD_CRIT' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING A FINITE COLLECTION OF MODELS. # MTB > # AND TWO PRIOR DISTRIBUTIONS # MTB > #--------------------------------------------------------------- # MTB > # INPUT: VALUES OF MODEL IN 'MODEL' AND PRIOR PROBABILITIES # MTB > # IN COLUMNS 'PRIOR1' AND 'PRIOR2' # MTB > # OUTPUT: POSTERIOR PROBABILITIES CORRESPONDING TO 2 PRIORS # MTB > # IN COLUMN 'POST1' AND 'POST2' # MTB > # BAYES FACTOR COMPARING TWO PRIORS # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 1 INPUT (number of successes, number of failures) DATA> 10 20 Row model prior1 prior2 POST1 POST2 1 0.20 0.05 0.20 0.015655 0.084360 2 0.22 0.05 0.20 0.024472 0.131873 3 0.24 0.10 0.20 0.069496 0.187250 4 0.26 0.25 0.15 0.226925 0.183429 5 0.28 0.25 0.10 0.275262 0.148333 6 0.30 0.15 0.05 0.187430 0.084169 7 0.32 0.10 0.05 0.133430 0.089878 8 0.34 0.05 0.05 0.067331 0.090708 TYPE 'y' AND RETURN FOR SUMMARIES: y FOR FIRST SET OF PRIOR PROBABILITIES: ------------------------------------- PRIOR MEAN OF MODELS: MEAN 0.274 POSTERIOR MEAN OF MODELS: MEAN 0.283087 FOR SECOND SET OF PRIOR PROBABILITIES: ------------------------------------- PRIOR MEAN OF MODELS: MEAN 0.247 POSTERIOR MEAN OF MODELS: MEAN 0.264901 -------------------------------------------------------- BAYES FACTOR IN FAVOR OF FIRST SET OF PRIOR PROBABILITIES: BAYES_F 1.34720 TYPE 'y' AND RETURN TO SEE PLOTS: y PLOT OF POSTERIOR PROBABILITIES FOR FIRST PRIOR: - 0.30+ - * PROB - * - 2 - 2 * 0.20+ 2 2 - 2 * 2 - * 2 2 - 2 * 3 * - 2 2 2 3 0.10+ 2 * 2 3 - * * 2 3 - 6 2 2 2 3 6 - 6 2 * 2 3 6 - 8 + 6 2 2 2 3 6 0.00+ + 9 3 * * 2 2 3 +---------+---------+---------+---------+---------+------MODELS 0.200 0.225 0.250 0.275 0.300 0.325 PLOT OF POSTERIOR PROBABILITIES FOR SECOND PRIOR: - * 0.180+ * 2 - * * PROB - * * - 2 * 2 - * * 2 2 0.120+ 2 * * * - 2 2 * 2 - 2 * 2 * * - 2 2 * * 2 2 3 2 - 3 * * * 2 3 3 3 0.060+ 3 2 2 2 * 3 2 3 - 3 2 * * 2 3 3 2 - 2 2 * * * 2 3 3 - 3 2 2 2 2 3 2 3 - 3 2 * * 2 3 3 2 0.000+ 2 * * * * 2 2 2 +---------+---------+---------+---------+---------+------MODELS 0.200 0.225 0.250 0.275 0.300 0.325 **************************** CHAPTER 10 ********************************* MTB > name c1 'prior_s' MTB > rand 1000 'prior_s'; SUBC> normal 6.9 .4. MTB > let 'prior_s'=exp('prior_s') MTB > exec 'mod_cont' MTB > ################################################################## MTB > # MACRO 'MOD_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING CONTINUOUS MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: SIMULATED VALUES OF PRIOR IN 'PRIOR_S' # MTB > # OUTPUT: SIMULATED VALUES FROM POSTERIOR IN 'POST_S' # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 6 INPUT (number of marked items, sample size, number marked in sample) DATA> 100 40 5 Input number of simulated values: DATA> 1000 Executing from file: lk_cap_n.MTB Each dot represents 5 points . . : ::::.:: ::::::::.. ::::::::::: .::::::::::: . .::::::::::::::.: :::::::::::::::::::. . ..::::::::::::::::::::.::::.....: .. . . . +---------+---------+---------+---------+---------+-------prior_s Each dot represents 7 points .. : .:: : :::::.: :::::::: . :::::::::: .::::::::::. .::::::::::::... . ::::::::::::::::.:........ . . +---------+---------+---------+---------+---------+-------POST_S 0 700 1400 2100 2800 3500 MTB > describe 'prior_s' 'post_s' Variable N Mean Median TrMean StDev SEMean prior_s 1000 1056.5 973.9 1022.2 453.8 14.4 POST_S 1000 909.58 860.38 889.20 308.11 9.74 Variable Min Max Q1 Q3 prior_s 281.2 3650.3 731.0 1271.1 POST_S 281.21 2516.90 681.46 1063.83 MTB > rand 900 'prior_s'; SUBC> beta 20.4 47.6. MTB > rand 100 c2; SUBC> unif 0 1. MTB > stack 'prior_s' c2 'prior_s' MTB > exec 'mod_cont' MTB > ################################################################## MTB > # MACRO 'MOD_CONT' # MTB > # -------------------------------------------------------------- # MTB > # INFERENCE USING CONTINUOUS MODELS. # MTB > #--------------------------------------------------------------- # MTB > # INPUT: SIMULATED VALUES OF PRIOR IN 'PRIOR_S' # MTB > # OUTPUT: SIMULATED VALUES FROM POSTERIOR IN 'POST_S' # MTB > ################################################################## INPUT THE NUMBER OF THE LIKELIHOOD: (1-Binomial P, 2-Normal M, 3-Poisson L, 4-Hypergeometric S, 5-Discrete Uniform N, 6-Capture/Recapture N, 7-Exponential M) DATA> 1 INPUT (number of successes, number of failures) DATA> 30 20 Input number of simulated values: DATA> 1000 Each dot represents 9 points . ::: :::: ::::::. ::::::: ::::::::: .:::::::::: .........::::::::::::...... ...... .............. +---------+---------+---------+---------+---------+-------prior_s Each dot represents 14 points . : : .: . :: .: :::. : . . ::: :::::: . ...::::..::: :::::: .::. +---------+---------+---------+---------+---------+-------POST_S 0.00 0.20 0.40 0.60 0.80 1.00 **************************** CHAPTER 11 ********************************* MTB > type 'logpost1.MTB' ################################################################## # MACRO 'LOGPOST1' # # -------------------------------------------------------------- # # DEFINITION OF THE LOGARITHM OF A ONE PARAMETER # # POSTERIOR DENSITY # ################################################################## ############################################### # binomial problem - cauchy prior # # x - logit of probability # ############################################### let k11=1 let k12=9 let 'f'=k11*'x'-(k11+k12)*log(1+exp('x'))-log(1+'x'**2) MTB > type 'logpost2.MTB' ################################################################## # MACRO 'LOGPOST2' # # -------------------------------------------------------------- # # DEFINITION OF THE LOGARITHM OF A TWO PARAMETER # # POSTERIOR DENSITY # ################################################################## ############################################### # 2 binomial problem # # t1 - difference in odds ratios # # t2 - sum of odds ratios # ############################################### set 'data' 1 10 3 12 let k21='data'(1) let k22='data'(2) let k23='data'(3) let k24='data'(4) let c201=('x'+'y')/2 let c202=('y'-'x')/2 let 'f'=k21*c201-(k21+k22)*log(1+exp(c201))+k23*c202-(k23+k24)*log(1+exp(c202)) MTB > exec 'laplace1' MTB > ################################################################## MTB > # MACRO 'LAPLACE1' # MTB > # -------------------------------------------------------------- # MTB > # SUMMARIZING A 1-PARAMETER POSTERIOR USING THE LAPLACE METHOD. # MTB > # DEFINITION OF LOG POSTERIOR IN MACRO 'LOGPOST1' # MTB > #--------------------------------------------------------------- # MTB > # INPUT: GUESS AT POSTERIOR MODE # MTB > # NUMBER OF ITERATIONS # MTB > # OUTPUT: CURRENT ESTIMATE AT MODE # MTB > # ASSOCIATED STANDARD DEVIATION AND ESTIMATE AT INTEGRAL# MTB > ################################################################## INPUT GUESS AT POSTERIOR MODE: DATA> 0 INPUT NUMBER OF ITERATIONS: DATA> 5 Row MODE STD LOG_INTG 1 -0.886723 0.470836 -6.76578 Row MODE STD LOG_INTG 1 -1.30571 0.673015 -4.394 Row MODE STD LOG_INTG 1 -1.41766 0.819856 -3.97834 Row MODE STD LOG_INTG 1 -1.42373 0.857813 -3.92344 Row MODE STD LOG_INTG 1 -1.42381 0.868545 -3.91098 MTB > exec 'laplace2' MTB > ################################################################## MTB > # MACRO 'LAPLACE2' # MTB > # -------------------------------------------------------------- # MTB > # SUMMARIZING A 2-PARAMETER POSTERIOR USING LAPLACE METHOD. # MTB > # DEFINITION OF LOG POSTERIOR IN MACRO 'LOGPOST2' # MTB > #--------------------------------------------------------------- # MTB > # INPUT: GUESS AT MODE # MTB > # NUMBER OF ITERATIONS # MTB > # OUTPUT: CURRENT ESTIMATE AT MODE # MTB > # ASSOCIATED COVARIANCE MATRIX AND INTEGRAL ESTIMATE # MTB > ################################################################## INPUT GUESS AT MODE: DATA> 1 1 INPUT NUMBER OF ITERATIONS: DATA> 5 Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -1.04684 0.85129 -3.43018 0.85129 0.19511 -22.3648 Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -0.927094 1.20075 -3.67809 1.20075 0.69814 -8.83597 Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -0.916306 1.23626 -3.68887 1.23626 0.702681 -8.71386 Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -0.916094 1.24168 -3.68866 1.24168 0.689256 -8.69772 Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -0.91617 1.28012 -3.68882 1.28012 0.799844 -8.66131 MTB > exec 'ad_quad1' MTB > ################################################################## MTB > # MACRO 'AD_QUAD1' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # SUMMARIZING A 1-PARAMETER POSTERIOR USING ADAPTIVE QUADRATURE.# MTB > # DEFINITION OF LOG POSTERIOR IN MACRO 'LOGPOST1' # MTB > #--------------------------------------------------------------- # MTB > # INPUT: MEAN AND STANDARD DEVIATION # MTB > # NUMBER OF ITERATIONS # MTB > # OUTPUT: FINAL GRID IN 'X', DENSITY VALUES IN 'F', # MTB > # WEIGHTS IN 'WX' # MTB > ################################################################## INPUT MEAN AND STANDARD DEVIATION: DATA> 0 1 INPUT NUMBER OF ITERATIONS: DATA> 5 - 0.0090+ * - density - - - 0.0060+ - * - - * - 0.0030+ - - * - - 0.0000+ * * * * * * --------+---------+---------+---------+---------+--------x -4.0 -2.0 0.0 2.0 4.0 - 0.0090+ * - density - - - 0.0060+ * - - - - 0.0030+ * - * - - - * 0.0000+ * * * * * --------+---------+---------+---------+---------+--------x -6.0 -4.0 -2.0 0.0 2.0 - 0.0090+ * - density - - - 0.0060+ * - - - - 0.0030+ - * * - - - * 0.0000+ * * * * * --------+---------+---------+---------+---------+--------x -6.0 -4.0 -2.0 0.0 2.0 - 0.0090+ * - density - - - 0.0060+ * - - - - 0.0030+ - * * - - - * 0.0000+ * * * * * --------+---------+---------+---------+---------+--------x -6.0 -4.0 -2.0 0.0 2.0 - 0.0090+ * - density - - - 0.0060+ * - - - - 0.0030+ - * * - - - * 0.0000+ * * * * * --------+---------+---------+---------+---------+--------x -6.0 -4.0 -2.0 0.0 2.0 Row MEAN STD LOG_INTG 1 -1.78599 0.99106 -3.86880 2 -1.80105 1.02136 -3.86742 3 -1.80249 1.02268 -3.86801 4 -1.80254 1.02275 -3.86802 5 -1.80254 1.02275 -3.86803 MTB > exec 'ad_quad2' MTB > ################################################################## MTB > # MACRO 'AD_QUAD2' # MTB > # (CHARACTER GRAPHICS VERSION) # MTB > # -------------------------------------------------------------- # MTB > # SUMMARIZING A 2-PARAMETER POSTERIOR USING ADAPTIVE QUADRATURE.# MTB > # DEFINITION OF LOG POSTERIOR IN MACRO 'LOGPOST2' # MTB > #--------------------------------------------------------------- # MTB > # INPUT: MEANS, STANDARD DEVIATIONS, AND COVARIANCE # MTB > # NUMBER OF ITERATIONS # MTB > # OUTPUT: FINAL GRID IN 'X','Y', DENSITY VALUES IN 'F', # MTB > ################################################################## INPUT MX, SX, MY, SY, COV: DATA> 0 1 0 1 0 INPUT NUMBER OF ITERATIONS: DATA> 5 y - - . . . . . . . . . . - 3.5+ . . . . . . . . . . - . . . . . . . . . . - - . . . . . . . . . . - . . . . . . . . . . 0.0+ - . . . . . . . . . . - . . . . / / . . . . - - . . . X X X / . . . -3.5+ . . X X X X / . . . - - . / X X X / . . . . - --------+---------+---------+---------+---------+--------x -4.0 -2.0 0.0 2.0 4.0 '0' < -1.4E-02 < '.' < 8.15E-03 < '/' < 3.06E-02 < 'X' y - . - . . - . . . . . 0.0+ . . . . . - . . . . . . . . . - . . . / . . . - . . . . X X / . . . - . . . X X X . . -4.0+ . . . / X X . . - . . / X X X / . . . - . . / X X / . . . - . . / / . . . . - . / . . . . . -8.0+ . . . . . - . . . . - . . - --+---------+---------+---------+---------+---------+----x -7.5 -5.0 -2.5 0.0 2.5 5.0 '0' < -1.1E-02 < '.' < 9.24E-03 < '/' < 2.91E-02 < 'X' - . y - . . . - . . . - . . . . . . 0.0+ . . . . . 2 . . - . . . . . . . 2 . - . . . . X 2 X . . 2 - . . . / X X X . . . - . . . 2 X / 2 . . -7.0+ 2 . / / / . . . . . - . 2 . . 2 . . . - . . . . . - . . . . . - . . . -14.0+ . - - --+---------+---------+---------+---------+---------+----x -9.0 -6.0 -3.0 0.0 3.0 6.0 '0' < -1.0E-02 < '.' < 5.83E-03 < '/' < 2.21E-02 < 'X' 7.0+ - . y - . . . - . . . . - . . . . . . 0.0+ . . . . . . . . - . . . . . . . 2 2 - . . . . X 2 / . . . - . . . / X X X . . . - . . . 2 X / 2 . . . -7.0+ . . / / / . . . . - 2 . . . . . . . - . 2 . . . . - . . . . . - . . -14.0+ . . - ----+---------+---------+---------+---------+---------+--x -9.0 -6.0 -3.0 0.0 3.0 6.0 '0' < -1.0E-02 < '.' < 5.68E-03 < '/' < 2.18E-02 < 'X' 7.0+ - . y - . . . - . . . . - . . . . . . 0.0+ . . . . . . . . - . . . . . . . 2 2 - . . . . X 2 / . . . - . . . / X X X . . . - . . . 2 X / 2 . . . -7.0+ . . / / / . . . . - 2 . . . . . . . - . 2 . . . . - . . . . . - . . -14.0+ . . - ----+---------+---------+---------+---------+---------+--x -9.0 -6.0 -3.0 0.0 3.0 6.0 '0' < -1.0E-02 < '.' < 5.68E-03 < '/' < 2.18E-02 < 'X' Row MN_1 STD_1 MN_2 STD_2 COVAR LOG_INTG 1 -0.98655 1.19910 -3.84017 0.96958 0.37392 -8.78444 2 -1.29675 1.46749 -4.33294 1.45820 1.18403 -8.60754 3 -1.30790 1.48900 -4.34742 1.48991 1.25498 -8.60532 4 -1.30811 1.48940 -4.34756 1.49036 1.25619 -8.60529 5 -1.30812 1.48940 -4.34756 1.49037 1.25621 -8.60529 MTB > exec 'metrop' MTB > ################################################################## MTB > # MACRO 'METROP' # MTB > # -------------------------------------------------------------- # MTB > # SIMULATING A 1-PARAMETER POSTERIOR USING THE METROPOLIS METHOD# MTB > # DEFINITION OF LOG POSTERIOR IN MACRO 'LOGPOST1' # MTB > #--------------------------------------------------------------- # MTB > # INPUT: STARTING VALUE # MTB > # NUMBER OF ITERATIONS # MTB > # OUTPUT: SIMULATED SAMPLE IN COLUMN 'POST_S' # MTB > ################################################################## INPUT NUMBER OF ITERATIONS: DATA> 500 INPUT SCALE OF NORMAL INCREMENT DENSITY: DATA> 1 INPUT STARTING VALUE: DATA> 0 Each dot represents 2 points : :. : : : :: : : . : . ::.:. : : . : ::..::::::.: :.::: . . :: ::::::::::::::::::: : : . . .:.:.:::::::::::::::::::::: ::. : : : ...:::::::::::::::::::::::::::::::::::.:..: +---------+---------+---------+---------+---------+-------POST_S -5.0 -4.0 -3.0 -2.0 -1.0 0.0 MTB > describe 'post_s' Variable N Mean Median TrMean StDev SEMean POST_S 500 -1.7252 -1.6693 -1.7012 0.9493 0.0425 Variable Min Max Q1 Q3 POST_S -4.7760 0.3120 -2.3659 -1.0699 MTB > exec 'gibbs' INPUT NUMBER OF ITERATIONS: DATA> 500 INPUT SCALES OF NORMAL INCREMENT DENSITIES: DATA> 1 1 INPUT STARTING VALUE: DATA> 0 0 MTB > plot 'post_y' 'post_x' - post_y - * ** - * **4 * 2 - 2* 2 *33* -2.5+ 4 * * **24 *2* 23423 * * 2 * - * * 2* *4222*3 42563532* *2** - * ** 23* 2 22*22252*3563332 * 3* * * - * *3 6 2*2* 5336**32 25434*54*5222 6* - * 2 2***23 *2*26*5333*3*24 3 *323 *2 -5.0+ *2* 4*222 *2 *3*2** 322**4 3*22*5*22* ** 2* ** - **2 *54 62334 4 *42 * 2 ** 32 2 * - 2 ** 2* 2 2 2 * 4* ** 3* - * ** ** ** ** - ** * * -7.5+ * * - 2 - ------+---------+---------+---------+---------+---------+post_x -3.6 -2.4 -1.2 0.0 1.2 2.4 MTB > dotplot 'post_x' 'post_y' . : . : .:. : : .::::: . :: .:::::: : . :: :::::::. : :: ::.::.:::::::: : . . ::::.::::::::::::::: : .:. : :::: ::::::::::::::::::::. : ::: : :::::::::::::::::::::::::: : . :::.::::::::::::::::::::::::::::: :. . .: : ::::::::::::::::::::::::::::::::: ::.:. ::::.::::::::::::::::::::::::::::::::: :::::... . . -----+---------+---------+---------+---------+---------+-post_x -3.6 -2.4 -1.2 0.0 1.2 2.4 Each dot represents 2 points . . . : : . .. :. . . .: :::. :: :: ::: :::.:::: :: . ::.:::::::::::: :: : . .:::::::::::::::::: :. .. :: :::::::::::::::::::..::: : . . . ::: ::.::::::::::::::::::::::::: :..: : -----+---------+---------+---------+---------+---------+-post_y -7.5 -6.0 -4.5 -3.0 -1.5 0.0 MTB > describe 'post_x' 'post_y' Variable N Mean Median TrMean StDev SEMean post_x 500 -1.1225 -0.9346 -1.1091 1.2503 0.0559 post_y 500 -4.1554 -4.1410 -4.1598 1.2163 0.0544 Variable Min Max Q1 Q3 post_x -3.9523 2.0727 -2.0531 -0.1369 post_y -7.7544 -0.7904 -5.0453 -3.3539