The Student’s t-test is the most commonly used statistical test for comparing two means or for comparing an observed mean with a known value. If more than two groups are two be compared, an analysis of variance (ANOVA) is used to compare means across groups.
PROC TTEST SYNTAX
PROC TTEST < options > ;
CLASS variable ;
PAIRED variables ;
BY variables ;
VAR variables ;
FREQ variable ;
WEIGHT variable ;
PROC TTEST OPTIONS
The following options can appear in the PROC TTEST statement:
ALPHA=pspecifies the confidence interval. The default value for ALPHA is 0.05;
CI=EQUALindicates an equal tailed confidence interval.
CI=UMPUindicates an interval based on the uniformly most powerful unbiased test of H0: σ = σ0;
COCHRANrequests the Cochran and Cox approximation of the probability level of the approximate t statistic for the unequal variances situation.
DATA= SAS-data-set specifies the name of the SAS data set for the procedure to use.
H0=mrequests tests against m instead of 0. By default H_0=0.
PROC TTEST Statements
The following statements are available in PROC TTEST:
CLASS statement is used to give the name of the classification or grouping variable that must accompany the PROC TTEST statement for two independent sample cases.
If the CLASS statement is used without the VAR statement all the numeric variables in the input data set excluding those appearing in the CLASS, BY, FREQ or weight statement. The CLASS variable should only have two levels and these can be either numeric or character variables.
CLASS statement should be omitted for the one sample or paired means comparisons.
PAIRED PairLists ;
PAIRED statements can be used only for Paired Comparisons. Pair Lists identifies the variables to be compared in paired comparisons. You can have one or more pair lists. You cannot use the CLASS and VAR statements with the PAIRED statement.
BY Statement :
BY Statement in PROC TTEST is used to obtain separate analyses on observations in groups defined by the BY variables. The input dataset needs to be sorted in order of the BY variables.
If your input data set is not sorted in ascending order you can specify the BY statement option with NOTSORTED or DESCENDING in the BY statement for the TTEST procedure. You can also create an INDEX on the BY variables.
The variables to be used in the analyses are named in the VAR statement. One-sample comparisons are conducted when the VAR statement is used without the CLASS statement, while group comparisons are conducted when the VAR statement is used with a CLASS statement.
If you omit the VAR statement, all the numeric variables available in the input dataset are used for analysis excluding the variables in the BY, CLASS, FREQ or WEIGHT statement.
You can use the VAR statements with one and two sample T Test but it cannot be used for PAIRED comparison.
FREQ- Statement :
FREQ statement Identifies a variable that contains the frequency of occurrence of each observation. PROC TTEST treats each observation as if it appears n times, where n is the value of the FREQ variable for the observation. If the value is not an integer, only the integer portion is used and if the frequency value is less than 1 or missing the observation is excluded from the analysis.
By default, each observation is assigned a frequency of 1 if you don’t specify the FREQ statement.
The FREQ statement cannot be used if the input data set contains summary statistics instead of the original observations.
WEIGHT variable ;
The WEIGHT statement weights each observation in the input dataset by the value of the WEIGHT variable. The values of the WEIGHT variable can be non-integral numbers nad they are not truncated.
If you don’t use the WEIGHT statement, by default each observation is assigned a weight of 1.If you have observations with negative, zero or missing values for the WEIGHT variable, then these variables are not used in the analysis.
The WEIGHT statement cannot be used if your input data set is a summary statistics and not raw data.
Performing a one-sample T-Test
In a one sample T-Test you obtain a random sample from some population and then compare the observed sample mean to some fixed value. A one sample T-Test is used to compare an observed mean with a known value. The purpose of one sample t-test is to determine if there is enough evidence to dispute the claim.
The hypotheis for a one sample t-test are :
H_0:\mu_0 = \mu_0 :The population mean is equal to a hypothesised value.
H_\alpha \mu \ne \mu_0 :The population mean is not equal to a hypotheised value.
The primary assumptions for one sample t-test are that the population from which the random sample is selected is normal. If the data are not normal then non-parametric test like sign test and signed-rank test is used.
In SAS, there are two procedures by which you can perform a one sample t-test.
- PROC UNIVARIATE
- PROC TTEST
In PROC UNIVARAITE, you can specify the value the value of \mu_0 using the option MU0=value.
PROC UNIVARIATE MU0=4;VAR LENGTH;RUN ;
would request a one-sample t-est of the null hypotheis that \mu_0=4
The second method in SAS for performing one sample ttest is using the PROC TTEST procedure.
PROC TTEST H0=4;VAR LENGTH;RUN;
A certain medical implant component is reported to be 4 cm in length by its manufacturer. To test the reliability of the manufacturer clay, a random sample of 20 of the component is collected.
For this example, we will compare the mean of the variable length in the clinic group for a pre-selected value of 4 and an alpha value of 0.1:
proc ttest data=data.clinic h0=4; var length; run;
- Variable – This is the name of variables used for the comparison.
- N – This is the number of observations used in calculating the t-test. (excluding missing values)
- Mean – This is the mean of the variable.
- Std Dev – This is the standard deviation of the variable.
- Std Err – This is the estimated standard deviation of the sample mean.
- Minimum – This is the minimum value in the input dataset.
- Maximum – This is the maximum value in the input dataset.
- 95% CL Mean – These are the lower and upper bound of the confidence interval for the mean.
- 95% CL Std Dev – These are the lower and upper bound of the confidence interval for standard deviation.
- DF – The degrees of freedom is the number of observations minus 1.
- t Value – This is the value of Student t-statistic. It is the ratio of the difference between the sample mean and the given number to the standard error of the mean.
- Since that the standard error of the mean measure the variability of the sample mean, the smaller the standard error of the mean, the more likely that our sample mean is close to the true population mean.
- Pr > |t| – The p-value is the two-tailed probability computed using the t distribution. It is the probability of observing a greater absolute value of t under the null hypothesis. For a one tailed test, half this probability.
- If the p-value is less than the pre-specified alpha level of 0.05 we will conclude that mean is statistically significantly different from zero. In this example, the p-value for length is greater than 0.5. So we conclude that the mean for length is not significantly different from 4 cm.
For the left graph, the blue curve is a normal curve based on the mean of 3.993 and standard deviation of 0.022. The red curve is a kernel density estimator.
In the Right graph, is a Q-Q plot that helps you to assess normality of the data.
Two sample t-test
The SAS PROC TTEST procedure is used to test for the equality of means for a two-sample (independent group) t-test.
The hypothesis for a two sample t-test are :
H_0: \mu_1 = \mu_2 :The population means of the two groups are equal.
H_\alpha \mu_1 \ne \mu_2 :The population means are not equal.
The underlying data for two-sample t-test are random samples and are independent. Another assumptions is that the populations are normally distributed with equal variances.
A biologist experimenting with plant growth designs an experiment in which 15 seeds are randomly assigned to one of the two fertilizers and the height of the resulting plant is measured after 2 weeks. She wants to know if one of the fertilizers provides more vertical growth than the other.
proc ttest data=data.grow; class brand; var height; run;
The first table is same as one sample TTest but informations are for each group and for the difference of mean.
The pooled estimator of variance is a weighted average of the two sample variances, with more weight given to the larger sample.
Satterthwaite is an alternative to the pooled-variance t-test and is used when the assumption that the two populations have equal variances seems unreasonable.
The second table gives the standard deviations again along with 95% confidence intervals for the means and standard deviations for each group.
The third table gives the results of the two t-test. The T-test based on a pooled estimate of the variance has a p-value of 0.0005 while for the Satterthwaite version, p=0.0008.
The fourth table gives the resukts of the F-Test for deciding whether the variance can be considered equal. The P-value for this test is 0.0388, so at 0.05 signifucance level we would consider that the variance are not equal and it appears that variance of Brand B is larger than that for Brand A.
The graphical representations show histogram, normal curves, kernel density estimators and box plot for both the groups. A Q-Q plot is also included in the output which is shown below.
Performing a Paired T-Test
Paired comparisons use the one sample process on the differences between the observations such as before and after situations where both observations are taken from the same or matched subjects.
Paired comparisons can be made between many pairs of variables with a single call to PROC TTEST Procedure.
Variables or lists of variables are separated by an asterisk (*) or a colon (:). The * tells SAS to compare each variable on the left with each variable on the right.
The : tells SAS to compare the first variable on the left to the first on the right, the second variable on the left and the second variable on the right, and so on.
It is important to use same number number of variables on the left and the right when the colon is used.
|PAIRED Statements Examples||The Comparisons|
|PAIRED AB CD||A-B and C-D|
|PAIRED (A B)*(C D);||A-C, A-D, B-C, and B-D|
|PAIRED (A B)*(C B);||A-C, A-B, and B-C|
|PAIRED (A1-A2)*(B1-B2);||A1-B1, A1-B2, A2-B1, and A2-B2|
|PAIRED (A1-A2):(B1-B2);||A1-B1 and A2-B2|
An input dataset contains the variables WBEFORE and WAFTER that represents before and after weight on a diet for 8 subjects. Perform a test to conclude if the average weight loss before and after the diet is not equal to 0.
proc ttest data=data.weight; paired wbefore * wafter; run;
From the above results, the P-value for this test is 0.0056, which provides evidence to reject the null hypothesis that is before and after weight loss is not 0.
The 95% confidence interval on the difference(7.23,22.26) indicates that the population mean weight loss could possibly lie between these two values. In either of the case, it appears that the mean weight loss is greater than 0.