Confidence interval for sample mean = Upper and lower confidence interval for sample standard deviation = Sample standard error =

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1 A Macro To Perform A T-Test For 2 Independent Samples Using Sufficient Statistics Lan-Feng Tsai, Edwards Lifesciences LLC, Irvine, California Abstract The T-test is a commonly used statistical test to compare the mean of one sample to a predetermined value, the means of paired samples, or the means of 2 independent samples. It is known that the test statistic for the T -test is based on the sample means, sample standard deviations, and sample sizes. Therefore, if only the summary statistics are known, and the raw data are unavailable, the result of the T -test can still be calculated. While SAS procedures require raw data to perform a T-test, a SAS macro to perform a T-test for 2 independent samples using sufficient statistics is proposed. The advantages of performing a T test using sufficient statistics are also discussed. Introduction A T -test can be performed using only summary statistics because the summary statistics are sufficient, consistent, and unbiased estimators for its normal model. The definition of a sufficient statistic is given as follows (Rice 1995): A statistic T(X 1,..., X,J is said to be sufficient for e if the conditional distribution of X1,..., Xn, given T=t, does not depend on 0 for any value oft. Confidence interval for sample mean = Upper and lower confidence interval for sample standard deviation = [ (n-1)s 2 2 a % Sample standard error = s.jn Sample mean difference = Pooled sample standard deviation (or standard deviation for sample mean difference): sp = 2 2 (n 1 -l)s 1 + (n 2 -l)s 2 n 1 +n 2-2 Pooled sample standard error = Calculation The purpose of this macro is to perform a T -test for 2 independent sample means using sufficient statistics (summary statistics). The theoretical details can be found in statistical textbooks (Arnold 1990, Rice 1995) and will not be discussed here. The following formulas are used to calculate the result of the T -test. Confidence interval for sample mean difference 68

2 Confidence interval for pooled sample standard deviation= F value for folded f statistic: f = P-value for folded f statistic = Degrees of freedom for equal variances = 2 [1-P(fmax(n1-1, n2-1), min(n1-1, n2-2) ::5 f)] Test statistic for equal variances: t_eq = P-value for equal variances = 2 P(t..1+n2-2 ::5 t_ eq) Degrees of freedom for unequal variances: df_uneq = Test statistic for unequal variances: t_uneq = Discussion Normality assumption One of the assumptions of the T -test is normality, and this assumption cannot be examined without the raw data. Therefore, one should keep in mind that the normality assumption might not be valid when comparing 2 independent sample means using summary statistics in a T -test. Advantages of performing T -tests using sufficient statistics Sometimes, statisticians obtain only summary statistics from clients, published litemture, or other sources. T -tests can still be performed keeping in mind the potential normality assumption violation. For example, we can compare the result of our product with the results of competitors products from published journals, companies websites, or advertisements. This macro can also be a handy tool when we would like to do a quick comparison of 2 sample means or to validate the results oft -tests with other statistical software. The Macro P-value for unequal variances = 2 P(t.Jr_uneq:St_uneq) Degrees of freedom for folded F statistic: df_f= The PROC TTEST in Version 8 (Appendix I) cannot perform a T -test using just summary statistics without a _STAT_ variable. Therefore, a STAT variable must be created along with the summary statistics that are to be in in the macro. The macro parameters MIN, MAX for both groups and the alpha level are not required. However, the numeric missing

3 values "." need to be entered to avoid confusion if MINs and MAXs are not used. The confidence intervals for the standard deviations of the groups are not calculated in the PROC TTEST. However, they can, in fact, be calculated using the formula provided above. This is expected to be solved in SAS Version 9. This macro creates a separate text file containing the confidence intervals for the standard deviations of the groups. A macro (Appendix 2) to perform a T -test using summary statistics in SAS Version 6 is also shown. Contact Information Lan-Feng Tsai One Edwards Way Irvine, CA Ian _feng_ tsai@edwards.com SAS is a registered trademark of SAS Institute Inc., Cary, NC, USA. Conclusion One sample T -tests and paired sample T tests can be performed using only summary statistics. More macros for such T-tests will be developed using summary statistics in the future. Acknowledgement The author would like to thank William Anderson PhD, Rita Kristy, Brian Ramos, and Felicia Ho for their generous comments. Reference Arnold, S. F., Mathematical Statistics (1990), Prentice-Hall, Inc., p.366, p.373. Rice, J. A., Mathematical Statistics and Data Analysis, Second Edition (1995), Duxbury Press, p.280, p.388. SAS/STAT Users Guide, Version 8, (1999), SAS Institute Inc. 70

4 Appendix 1. Version 8 SAS code: ttest8 macro: Perform V8 proc ttest using summary ; statistics ; Position parameters gl: sample name of group 1 nl: sample size of group 1 m1: sample mean of group 1 sl: sample standard error of group 1 il: sample minimum of group 1 xl: sample maximum of group l g2: sample name of group 2 n2: sample size of group 2 m2: sample mean of group 2 s2: sample standard error of group 2 i2: sample minimum of group 2 x2: sample maximum of group 2 alpha: alpha level (default is 0.05) ; ; ; ; ; ; ; ; ; Note: il, xl, i2, x2 are not required, enter values or for missing. Written by Lan-Feng Tsai %macro ttest8(g1, n1, ml, s1, il, x1, g2, n2, m2, s2, i2, x2, alpha); data sumstat; %let len=%sysfunc(max(%length(&gl), %length(&g2))); length group $&len. stat $4.; %do i=1 %to 2; - - group="&&g&i"; sumstat=&&n&i; _stat_=n; out; group="&&g&i"; sumstat=&&m&i; _stat_=mean; out; proc print; group="&&g&i"; sumstat=&&s&i; stat_=std; out; group="&&g&i"; sumstat=&&i&i; _stat_=min; out; group="&&g&i"; sumstat=&&x&i; _stat_=max; out; proc ttest %if &alpha ne %then %do; alpha=&alpha ; class group; var sumstat; data null ; file ttestmacro_v8.txt; gl="&g1"; n1=&nl; s1=&s1; g2="&g2"; n2=&n2; s2=&s2; %if &alpha ne %then %do; lcll=sqrt(((n1-1)*sl**2)/cinv((1-&alpha/2), n1-1)); ucll=sqrt(((nl-l)*s1**2)/cinv(&alpha/2, n1-1)); lcl2=sqrt(((n2-1)*s2**2)/cinv((1-&alpha/2), n2-1)); ucl2=sqrt(((n2-l}*s2**2)/cinv(&alpha/2, n2-1)); %else %do; 1cll=sqrt(((n1-l}*s1**2)/cinv(0.975, nl-1}); ucll=sqrt(((nl-l}*s1**2}/cinv(0.025, n1-1}); lcl2=sqrt(((n2-1}*s2**2)/cinv(0.975, n2-l)); ucl2=sqrt(((n2-l}*s2**2)/cinv(0.025, n2-1)); 71

5 @21 LCL UCL ucl2; %mend ttestb; 2. Version 6 SAS code: *; ttest6 macro: gives similar out as proc ttest position parameters V6 using sufficient statistics mgl: name of group 1 mn1: sample size of group 1 mml: sample mean of group 1 ms1: sample standard error of group 1 mg2: name of group 2 mn2: sample size of group 2 mm2: sample mean of group 2 ms2: sample standard error of group 2 Note: specify out file out in a written by : Lan-Feng Tsai FILENAME statement. ; ; ; ; ; ; * %macro ttest6(mgl, mnl, mml, msl, mg2, mn2, mm2, ms2); data null ; file ttestmacro V6.txt; attrib gl g2 for;at=$8. nl n2 format=s. m1 m2 sl s2 t_uneq t_eq f format=8.2 df_uneq df_eq format~s.l f_p t_p_uneq t_p_eq format=8.4; gl="&mgl"; nl=&mnl; ml=&mml; sl=&msl; g2="&mg2"; n2=&mn2; m2=&mm2; s2=&ms2; vl=sl**2; v2=s2**2; f~ax(of vl, v2)/min(of vl, v2); dfl=nl-1; df2=n2-1; dfmax=max(of dfl, df2); dfmin=min(of dfl, df2); f_p=2*(1-probf(f, dfmax, dfmin)); 2-sided ; v_pool=((nl-l)*v1+(n2-l)*v2)/(nl+n2-2); t uneq=(ml-m2)/sqrt(vl/nl+v2/n2); t=eq~(ml-m2)/sqrt(v_pool*(l/nl+l/n2)); df uneq=(v1/n1+v2/n2)**2/((v1/n1)**2/(nl-l)+(v2/n2)**2/(n2-1)); df=eq=nl+n2-2; t_p_uneq=2*(1-probt(abs(t_uneq), df_uneq)); t_p_eq=2*(1-probt(abs(t_eq), df_eq)); Std Err; s2; ; Prob> IT I ; df For HO: Variances are equal, F" = f +5 DF = ( dfmax +(-1), dfmin +(-1) ) +5 Prob>F 1 f_p; %mend ttest6; ; 72