1          OPTIONS NONOTES NOSTIMER NOSOURCE NOSYNTAXCHECK;
 70         
 71         title 'The Population Variation Method for Selecting Sample Size';
 72         
 73         /***********************  popvar.sas ****************************
 74         * This code writes on the log file.                             *
 75         /***************************************************************/
 76         
 77         /*****************************************************************
 78         Given a population effect size A, what sample size is required
 79         to detect the effect with a given probability?
 80         
 81         Suppose we are planning a 2x3x4 analysis of covariance,
 82         with two covariates, and factors named A, B and C. We
 83         are setting it up as a regression model, with one dummy
 84         variable for A, 2 dummy variables for B, and 3 for C.
 85         Interactions are represented by product terms, and there
 86         are 2 products for the AxB interaction, 3 for AxC, 6 for
 87         BxC, and 1*2*3 = 6 for AxBxC. The regression coefficients
 88         for these plus two for the covariates and one for the
 89         intercept give us p = 26. The null hypothesis is that of no
 90         BxC interaction, so s = 6. The "other effects in the
 91         model" for which we are "controlling" are represented
 92         by 2 covariates and 17 dummy variables and products of
 93         dummy variables.
 94         
 95         What sample size is required for a power of 0.80 if the BxC
 96         interaction explains 10% of the remaining variation IN THE
 97         POPULATION?  The sample variation method yielded n = 144 for
 98         this problem.
 99         *****************************************************************/
 100        
 101                            /*********************************************/
 102        data fpower1;       /* Replace alpha, s, p, and wantpow below    */
 103             alpha = 0.05;  /* Significance level                        */
 104             s = 6;         /* Numerator df = # IVs being tested         */
 105             p = 26;        /* There are p beta parameters               */
 106             A = 0.10;      /* POPULATION effect size                    */
 107             wantpow = .80; /* Find n to yield this power.               */
 108                            /*********************************************/
 109             power = 0; n = p; oneminus = 1-alpha; /* Initializing ... */
 110             do until (power >= wantpow);
 111                n = n+1 ;
 112                ncp = (n-p)*A/(1-A);
 113                df2 = n-p;
 114                power = 1-probf(finv(oneminus,s,df2),s,df2,ncp);
 115             end;
 116             put ' *********************************************************';
 117             put '   ';
 118             put '   For a multiple regression model with ' p 'betas, ';
 119             put '   testing ' s 'explanatory variables using alpha = ' alpha ',';
 120             put '   a sample size of ' n 'is needed';
 121             put '   in order to have probability ' wantpow 'of rejecting H0';
 122             put '   for a POPULATION  effect of size A = ' A ;
 123             put '   ';
 124             put ' *********************************************************';
 125        run;
 
  *********************************************************
    For a multiple regression model with 26 betas, 
    testing 6 explanatory variables using alpha = 0.05 ,
    a sample size of 155 is needed
    in order to have probability 0.8 of rejecting H0
    for a POPULATION  effect of size A = 0.1
  *********************************************************
 NOTE: The data set WORK.FPOWER1 has 1 observations and 10 variables.
 NOTE: DATA statement used (Total process time):
       real time           0.00 seconds
       user cpu time       0.00 seconds
       system cpu time     0.00 seconds
       memory              734.62k
       OS Memory           28072.00k
       Timestamp           04/01/2020 08:00:02 AM
       Step Count                        70  Switch Count  2
       Page Faults                       0
       Page Reclaims                     169
       Page Swaps                        0
       Voluntary Context Switches        10
       Involuntary Context Switches      0
       Block Input Operations            0
       Block Output Operations           264
       
 
 126        
 127        /******************************************************************/
 128        /* Given sample size, what effect size (population A) is required */
 129        /* to have a specified power?                                     */
 130        /******************************************************************/
 131        
 132        /* This example uses the same design as above. Suppose we did have the
 133           n = 144 for a = 0.10 located by the sample variation method. What
 134           POPULATION effect size would be necessary for this sample size to
 135           yield a power of 0.80?  */
 136        
 137                            /*********************************************/
 138        data fpower2;        /* Replace alpha, s, n, p, and wantpow below */
 139             alpha = 0.05;  /* Significance level                        */
 140             s = 6;         /* Numerator df = # IVs being tested         */
 141             n = 144;       /* Sample size                               */
 142             p = 26;        /* There are p beta parameters               */
 143             wantpow = .80; /* Find effect size A to yield this power.   */
 144                            /*********************************************/
 145             df2 = n-p; oneminus = 1 - alpha;
 146             critval = finv(oneminus,s,df2);
 147           /* Initializing ... */  A = 0;
 148           do until (power ge wantpow);
 149              A = A + .001 ;
 150              ncp = (n-p)*A/(1-A);
 151              power = 1-probf(critval,s,df2,ncp);
 152             end;
 153           put ' ******************************************************';
 154           put ' ';
 155           put '  For a multiple regression model with ' p 'betas, ';
 156           put '  testing ' s ' explanatory at significance level ';
 157           put '  alpha = ' alpha ' controlling for the other variables,';
 158           put '  and a sample size of ' n', the variables need to explain';
 159           put '  A = ' A ' of the remaining POPULATION variation to have a';
 160           put '  probability of '  wantpow  'of being significant';
 161           put '   ';
 162           put ' *******************************************************';
 163        run;
 
  ******************************************************
   For a multiple regression model with 26 betas, 
   testing 6  explanatory at significance level 
   alpha = 0.05  controlling for the other variables,
   and a sample size of 144 , the variables need to explain
   A = 0.109  of the remaining POPULATION variation to have a
   probability of 0.8 of being significant
  *******************************************************
 NOTE: The data set WORK.FPOWER2 has 1 observations and 11 variables.
 NOTE: DATA statement used (Total process time):
       real time           0.00 seconds
       user cpu time       0.00 seconds
       system cpu time     0.00 seconds
       memory              701.40k
       OS Memory           28072.00k
       Timestamp           04/01/2020 08:00:02 AM
       Step Count                        71  Switch Count  2
       Page Faults                       0
       Page Reclaims                     95
       Page Swaps                        0
       Voluntary Context Switches        9
       Involuntary Context Switches      0
       Block Input Operations            0
       Block Output Operations           264
       
 
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 167        OPTIONS NONOTES NOSTIMER NOSOURCE NOSYNTAXCHECK;
 178