Now consider a situation in which we compare the difference between means of samples selected from two populations.
Z AND t STATISTICS FOR TWO INDEPENDENT SAMPLES
Now consider a situation in which we compare the
difference between means of samples selected from two populations. In a
clinical trial, we could be comparing the mean of a variable (commonly referred
to as an endpoint) for a control group to the corresponding mean for a
treatment group.
First assume that both groups have normally
distributed observations with known and possibly different variances σt and σc for the treatment and control groups, respectively. Assume that the
sample size for the treatment group is nt
and for the control group is nc.
Also assume that the means are μt and μc for the treatment and control groups, respectively.
Let us select two samples independently from the two groups (treatment and control) and compute the means of the samples. We denote the means of the samples from the control and treatment groups, t and c, respectively. The difference between the sample means t c comes from a normal distribution with mean μt – μc, variance (σ2t/nt) + (σ2c/nc), and standard error for t – c equal to √(σt /nt) + (σc/nc). The Z transformation of t – c is defined as
which has a standard normal distribution. Here is
an interesting statistical observation: Even though we are finding the
difference between two sample means, the variance of the distribution of their
differences is equal to the sum of the two squared standard errors associated
with each of the individual sample means. The standard errors of the treatment
and control groups are calculated by dividing the population variance of each
group by the respective sample size of each independently selected sample.
As demonstrated in Section 8.6, the Z transformation, which employs the addition
of the error variances of the two means, enables us to obtain confidence
intervals for the difference between the means. In the special case where we
can assume that σ2t = σ c2 = σ2, the Z formula reduces to
The term σ2 is referred to as the common variance. Since P{[–1.96 ≤ Z ≤ 1.96] = 0.95, we find after algebraic manipulation that
is a 95% confidence interval for μt – μc.
In practice, the population variances of the treatment
and control groups are unknown; if the two variances can be assumed to be equal,
we can calculate an estimate of the common variance σ2, called the pooled estimate. Let S 2t and Sc2
be the sample estimates of the variance for the treatment and control groups,
respectively. The pooled variance estimate Sp2
is then given by the formula
The corresponding t statistic is
This formula is obtained by replacing the common s in the formula above for Z
with the pooled estimate Sp.
The resulting statistic has Student’s t
distribution with nt + nc – 2 degrees of freedom.
We will use this formula in Section 8.7 to obtain a confidence interval for the mean difference based on this t statistic when the popu-lation
variances can be assumed to be equal.
Although not covered in this text, the hypothesis
of equal variances can be tested by an F
test similar to the F tests that are
used in the analysis of variance (discussed in Chapter 13). If the F test indicates that the variances are
different, then one should use a statistic based on the assumption of unequal
variances.
This problem with unequal and unknown variances is
called the Behrens–Fisher problem. Let “k”
denote the test statistic that is commonly used in the Behrens– Fisher problem.
The test statistic k does not have a t distribution, but it can be
ap-proximated by a t distribution
with a degrees of freedom parameter that is not nec-essarily an integer. The statistic
k is obtained by replacing the Z statistic in the un-equal variance
case as given below:
where S2t and S c2
are the sample estimates of variance for the treatment and control groups,
respectively.
We use a t
distribution with v degrees of freedom to approximate the distribution
of k. The degrees of freedom v are
This is the formula we use for confidence intervals
in Section 8.7 when the vari-ances are assumed to be unequal and also for
hypothesis testing under the same as-sumptions (not covered in the text).
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