The **ttest** procedure performs t-tests for one sample, two samples and
paired observations. The single-sample t-test compares the mean of the
sample to a given number (which you supply). The dependent-sample t-test
compares the difference in the means from the two variables to a
given number (usually 0), while taking into account the fact that the scores are
not independent. The independent samples t-test compares the difference in
the means from the two groups to a given value (usually 0). In other
words, it tests whether the difference in the means is 0. In
our examples, we will use the hsb2
data set.

## Single sample t-test

For this example, we will compare the mean of the variable **write** with
a pre-selected value of 50. In practice, the value against which the mean
is compared should be based on theoretical considerations and/or previous
research.

proc ttest data="D:hsb2" H0=50; var write; run;

The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable N Mean Mean Mean Std Dev Std Dev Std Dev Std Err write 200 51.453 52.775 54.097 8.6318 9.4786 10.511 0.6702 T-Tests Variable DF t Value Pr > |t| write 199 4.14 <.0001

## Summary statistics

Statistics Lower CL Upper CL Lower CL Upper CL VariableN^{a}Mean^{b}Mean^{c}Mean^{d}Std Dev^{c}Std Dev^{e}Std Dev^{f}Std Err^{e}write 200 51.453 52.775 54.097 8.6318 9.4786 10.511 0.6702^{g}

a. **Variable** – This is the list of variables. Each variable
that was listed on the **var** statement will have its own line in this part
of the output.

b. **N** – This is the number of valid (i.e., non-missing)
observations used in calculating the t-test.

c. **Lower CL Mean **and** Upper CL Mean** – These are the lower
and upper bounds of the confidence interval for the mean. A confidence interval
for the mean specifies a range of values within which the unknown population
parameter, in this case the mean, may lie. It is given by

where *s*
is the sample deviation of the observations and N is the number of valid
observations. The t-value in the formula can be computed or found in any
statistics book with the degree of freedom being N-1 and the p-value being 1-*alpha*/2,
where *alpha* is the confidence level and by default is .95. If we
drew 200 random samples, then about 190 (200*.95) times, the confidence interval
would capture the parameter mean of the population.

d. **Mean** – This is the
mean of the variable.

e. **Lower CL Std Dev** and **Upper CL Std Dev** – Those are the
lower and upper bound of the confidence interval for the standard deviation. A
confidence interval for the standard deviation specifies a range of values
within which the unknown parameter, in this case, the standard deviation, may
lie. The computation of the confidence interval is based on a chi-square
distribution and is given by the following formula

where **S**^{2} is the estimated variance of the variable and
alpha is the confidence level. If we drew 200 random samples, then about 190
(200*.95) of times, the confidence interval would capture the parameter standard
deviation of the population.

f. **Std Dev** – This is the standard deviation of the variable.

g. **Std Err** – This is the estimated standard deviation of the
sample mean. If we drew repeated samples of size 200, we would expect the
standard deviation of the sample means to be close to the standard error.
The standard deviation of the distribution of sample mean is estimated as the
standard deviation of the sample divided by the square root of sample size.
This provides a measure of the variability of the sample mean. The Central
Limit Theorem tells us that the sample means are approximately normally
distributed when the sample size is 30 or greater.

## Test statistics

The single sample t-test tests the null hypothesis that the population mean
is equal to the given number specified using the option **H0= **. The
default value in SAS for H0 is 0. It calculates the t-statistic and its
p-value for the null hypothesis under the assumption that the sample comes from
an approximately normal distribution. If the p-value associated with the t-test
is small (usually set at p < 0.05), there is evidence that the mean is different
from the hypothesized value. If the p-value associated with the t-test is
not small (p > 0.05), then the null hypothesis is not rejected, and you conclude
that the mean is not different from the hypothesized value.

In our example, the t-value for variable **write** is 4.14 with 199
degrees of freedom. The corresponding p-value is .0001, which is less than
0.05. We conclude that the mean of variable **write** is different from
50.

T-Tests VariableDF^{a}t Value^{h}Pr > |t|^{i}write 199 4.14 <.0001^{j}

a. **Variable** – This is the list of variables. Each variable
that was listed on the **var** statement will have its own line in this part
of the output.
If a **var**
statement is not specified, **proc ttest** will conduct a t-test on all
numerical variables in the dataset.

h. **DF** – The
degrees of freedom for the single sample t-test is simply the number of valid
observations minus 1. We loose one degree of
freedom because we have estimated the mean from the sample. We have used
some of the information from the data to estimate the mean; therefore, it is not
available to use for the test and the degrees of freedom accounts for this.

i. **t Value** – This is the 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. This is illustrated by the following three figures.

All three cases the difference between the population means are the same. But with large variability of sample means, two populations overlap a great deal. Therefore, the difference may well come by chance. On the other hand, with small variability, the difference is more clear. The smaller the standard error of the mean, the larger the magnitude of the t-value. Therefore, the smaller the p-value. The t-value takes into account of this fact.

j. **Pr > |t|** – The p-value is the two-tailed
probability computed using t distribution. It is the probability of
observing a
greater absolute value of t under the null hypothesis. For a one-tailed
test, halve this probability. If p-value is less than the pre-specified
alpha level (usually .05 or .01) we
will conclude that mean is statistically significantly different from zero. For example,
the p-value for **write** is smaller than 0.05. So we conclude that the
mean for **write** is significantly different from 50.

## Dependent group t-test

A dependent group t-test is used when the observations are not independent of
one another. In the example below, the same students took both the writing
and the reading test. Hence, you would expect there to be a relationship
between the scores provided by each student. The dependent group t-test
accounts for this. In the example below, the t-value for the difference
between the variables **write** and **read** is 0.87 with 199
degrees of freedom, and the corresponding p-value is .3868. This is greater
than our pre-specified alpha level, 0.05. We conclude that the difference between the variables **
write** and **read** is not statistically significantly different from 0.
In other words, the means for **write** and **read** are not statistically
significantly different from one another.

proc ttest data="D:hsb2"; paired write*read; run;

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL Upper CL Difference N Mean Mean Mean Std Dev Std Dev Std Dev Std Err

write - read 200 -0.694 0.545 1.7841 8.0928 8.8867 9.8546 0.6284

T-Tests

Difference DF t Value Pr > |t|

write - read 199 0.87 0.3868

## Summary statistics

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL Upper CL DifferenceN^{a}Mean^{b}Mean^{c}Mean^{d}Std Dev^{c}Std Dev^{e}Std Dev^{f}Std Err^{e}^{g}

write - read 200 -0.694 0.545 1.7841 8.0928 8.8867 9.8546 0.6284

a. **Difference** – This is the list of variables.

b. **N** – This is the number of valid (i.e., non-missing)
observations used in calculating the t-test.

c. **Lower CL Mean **and** Upper CL Mean** – These are the lower
and upper bounds of the confidence interval for the mean. A confidence
interval for the mean specifies a range of values within which the unknown
population parameter, in this case the mean, may lie. It is given by

where *s*
is the sample deviation of the observations and N is the number of valid
observations. The t-value in the formula can be computed or found in any
statistics book with the degree of freedom being N-1 and the p-value being 1-*alpha*/2,
where *alpha* is the confidence level and by default is .95. If we
drew 200 random samples, then about 190 (200*.95) times, the confidence interval
would capture the parameter mean of the population.

d. **Mean** – This is the
mean of the variable.

e. **Lower CL Std Dev** and **Upper CL Std Dev** – Those are the
lower and upper bound of the confidence interval for the standard deviation. A
confidence interval for the standard deviation specifies a range of values
within which the unknown parameter, in this case, the standard deviation, may
lie. The computation of the confidence interval is based on a chi-square
distribution and is given by the following formula

where **S**^{2} is the estimated variance of the variable and
alpha is the confidence level. If we drew 200 random samples, then about 190
(200*.95) of times, the confidence interval would capture the parameter standard
deviation of the population.

f. **Std Dev** – This is the standard deviation of the variable.

g. **Std Err** – This is the estimated standard deviation of the
sample mean. If we drew repeated samples of size 200, we would expect the
standard deviation of the sample means to be close to the standard error.
The standard deviation of the distribution of sample mean is estimated as the
standard deviation of the sample divided by the square root of sample size.
This provides a measure of the variability of the sample mean. The Central
Limit Theorem tells us that the sample means are approximately normally
distributed when the sample size is 30 or greater.

## Test statistics

T-Tests

DifferenceDF^{h}t Value^{i}Pr > |t|^{j}^{k}

write - read 199 0.87 0.3868

h. **Difference** – The t-test for dependent groups is to form a
single random sample of the paired difference. Therefore, essentially it is a
simple random sample test. The interpretation for t-value and p-value is the
same as for the case of simple random sample.

i. **DF** – The degrees of freedom for the paired observations is
simply the number of observations minus 1. This is because the test is conducted
on the one sample of the paired differences.

j. **t Value** – This is the t-statistic. It is the ratio of
the mean of the difference in means to the standard error of the difference
(.545/.6284).

k. **Pr > |t|** – The p-value is the two-tailed probability computed
using t distribution. It is the probability of observing a greater absolute value of
t under the null hypothesis. For a one-tailed test, halve this
probability. If p-value is less than our pre-specified alpha level,
usually 0.05, we will conclude that
the difference is significantly from zero. For example, the p-value for
the difference between write and read is greater than 0.05, so we conclude that
the difference in means is not statistically significantly different from 0.

## Independent group t-test

This t-test is designed to compare means of same variable between two groups.
In our example, we compare the mean writing score between the group of
female students and the group of male students. Ideally, these subjects are
randomly selected from a larger population of subjects. Depending on if we
assume that the variances for both populations are the same or not, the standard
error of the mean of the difference between the groups and the degree of freedom
are computed differently. That yields two possible different t-statistic and two
different p-values. When using the t-test for comparing independent groups, we
need to test the hypothesis on equal variance and this is a part of the output
that **proc ttest** produces. The interpretation for p-value is the same as
in other type of t-tests.

proc ttest data="D:hsb2"; class female; var write; run;

The TTEST Procedure

Statistics

Lower CL Upper CL Lower CL Upper CL Variable female N Mean Mean Mean Std Dev Std Dev Std Dev Std Err

write 0 91 47.975 50.121 52.267 8.9947 10.305 12.066 1.0803 write 1 109 53.447 54.991 56.535 7.1786 8.1337 9.3843 0.7791 write Diff (1-2) -7.442 -4.87 -2.298 8.3622 9.1846 10.188 1.3042

T-Tests

Variable Method Variances DF t Value Pr > |t|

write Pooled Equal 198 -3.73 0.0002 write Satterthwaite Unequal 170 -3.66 0.0003

Equality of Variances

Variable Method Num DF Den DF F Value Pr > F

write Folded F 90 108 1.61 0.0187

## Summary statistics

Statistics

Lower CL Upper CL Lower CL Upper CL Variablefemale^{a}N^{b}Mean^{c}Mean^{d}Mean^{e}Std Dev^{d}Std Dev^{f}Std Dev^{g}Std Err^{f}^{h}

write 0 91 47.975 50.121 52.267 8.9947 10.305 12.066 1.0803 write 1 109 53.447 54.991 56.535 7.1786 8.1337 9.3843 0.7791 write Diff (1-2) -7.442 -4.87 -2.298 8.3622 9.1846 10.188 1.3042

a. **Variable** – This column lists the dependent variable(s).
In our example, the dependent variable is **write**.

b. **female** –
This column gives
values of the class variable, in our case **female**. This variable is
necessary for doing the independent group t-test and is specified by **class**
statement.

c. **N** – This is the number of valid (i.e., non-missing)
observations in each group defined by the variable listed on the **class**
statement (often called the independent variable).

d. **Lower CL Mean** and **Upper CL Mean** – These are the lower
and upper confidence limits of the mean. By default, they are 95%
confidence limits.

e. **Mean** – This is the mean of the dependent variable for each
level of the independent variable. On the last line the difference between
the means is given.

f. **Lower CL Std Dev** and **Upper LC Std Dev** – These are the
lower and upper 95% confidence limits for the standard deviation for the
dependent variable for each level of the independent variable.

g. **Std Dev** – This is the standard deviation of the dependent
variable for each of the levels of the independent variable. On the last
line the standard deviation for the difference is given.

h. **Std Err** – This is the standard error of the mean.

## Test statistics

T-Tests

VariableMethod^{a}Variances^{i}DF^{j}t Value^{k}Pr > |t|^{l}^{m}

write Pooled Equal 198 -3.73 0.0002 write Satterthwaite Unequal 170 -3.66 0.0003

Equality of Variances

VariableMethod^{a}Num DF^{i}Den DF^{n}F Value^{n}Pr > F^{o}^{p}

write Folded F 90 108 1.61 0.0187

a. **Variable** – This column lists the dependent variable(s). In
our example, the dependent variable is **write**.

i. **Method** – This column specifies the method for computing the
standard error of the difference of the means. The method of computing
this value is based on the assumption regarding the
variances of the two groups. If we assume that the two populations have the same variance,
then the first method, called pooled variance estimator, is used. Otherwise, when
the variances are not assumed to be equal, the Satterthwaite’s method is used.

j. **Variances** – The pooled estimator of variance is a weighted
average of the two sample variances, with more weight given to the larger sample
and is defined to be

s^{2} =
((n1-1)s1+(n2-1)s2)/(n1+n2-2),

where s1 and s2
are the sample variances and n1 and n2 are the sample sizes for the two groups.
the This is called pooled variance. The standard error of the mean of the
difference is the pooled variance adjusted by the sample sizes. It is defined to
be the square root of the product of pooled variance and (1/n1+1/n2). In our
example, n1=109, n2=91. The pooled variance = 108*8.1337^{2}+90*10.305^{2}/198=84.355.
It follows that the standard error of the mean of the difference =
sqrt(84.355*(1/109+1/91))=1.304. This yields our t-statistic to be
-4.87/1.304=-3.734.

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. It provides a t statistic that asymptotically (that is, as the sample sizes become large) approaches a t distribution, allowing for an approximate t test to be calculated when the population variances are not equal.

k. **DF** – The degrees of freedom for the paired observations is
simply the number of observations minus 2. We use one degree of freedom for
estimating the mean of each group, and because there are two groups, we use two
degrees of freedom.

l. **t Value** – This t-test is designed to compare means between
two groups of the same
variable such as in our example, we compare the mean writing
score between the group of female students and the group of male students.
Depending on if we assume that the variances for both
populations are the same or not, the standard error of the mean of the
difference between the groups and the degrees of freedom are computed
differently. That yields two possible different t-statistic and two
different p-values. When using the t-test for comparing independent
groups, you need to look at the variances for the two groups. As long as the two
variances are close (one is not more than two or three times the other), go with
the equal variances test. The interpretation for the p-value is the same as in
other types of t-tests.

m. **Pr > |t|** – The p-value is the two-tailed probability
computed using the t distribution. It is the probability of observing a t-value of
equal or greater absolute value under the null hypothesis. For a
one-tailed test, halve this probability. If the p-value is less than our
pre-specified alpha level, usually 0.05, we will conclude that the difference is significantly different from zero.
For example, the p-value for the difference between females and males is less
than 0.05, so we conclude that the difference in means is statistically
significantly different from 0.

n. **Num DF** and **Den DF** – The F distribution is the ratio of
two estimates of variances. Therefore it has two parameters, the degrees of
freedom of the numerator and the degrees of freedom of the denominator. In SAS
convention, the numerator corresponds to the sample with larger variance and the
denominator corresponds to the sample with smaller variance. In our example,
the male students group ( **female**=0) has variance of 10.305^2 (the standard
deviation squared) and for the female
students the variance is 8.1337^2. Therefore, the degrees of freedom for the numerator is
91-1=90 and the degrees of freedom for the denominator 109-1=108.

o. **F Value** – SAS labels the F statistic not F, but F’, for a
specific reason. The test statistic of the two-sample F test is a ratio of
sample variances, F = s_{1}^{2}/s_{2}^{2} where
it is completely arbitrary which sample is labeled sample 1 and which is
labeled sample 2. SAS’s convention is to put the **larger** sample variance in
the numerator and the smaller one in the denominator. This is called the **
folded** F-statistic,

**F’ = max(s _{1}^{2},s_{2}^{2})/min(s_{1}^{2},s_{2}^{2})**

which will always be greater than 1. Consequently, the F test rejects the null hypothesis only for large values of F’. In this case, we get 10.305^2 / 8.1337^2 = 1.605165, which SAS rounds to 1.61.

p. **Pr > F** –
This is the
two-tailed significance probability. In our example, the probability is less
than 0.05. So there is evidence that the variances for the two groups, female
students and male students, are different. Therefore, we may want to use the
second method (Satterthwaite variance estimator) for our t-test.