The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

6.5 Paired t-Test 77

This will usually lead to a smaller variance for the difference of the

two means, such as in crossover trials or the change in test scores in a

particular subject before and after an intervention.

Steps for the paired t - test.

1. 
   

   Form the paired differences d i = X Ti − X Ci for i = 1, 2,... , n,
   where i is the index for the paired variables (e.g., patients)

   
   


2. 
   

   State the null hypothesis H 0 : μ (^) T = μ (^) C versus the alternative
   H 1 : μ (^) T ≠ μ (^) C (or equivalently H 0 : μ (^) T − μ (^) C = 0 versus the alterna-
   tive H 1 : μ (^) T − μ (^) C ≠ 0)

   
   


3. 
   

   Choose a signifi cance level α (often α = 0.01, 0.05, or 0.10).

   
   


4. 
   

   Determine the critical region: the region that has values of the
   test statistic t in the upper α /2 or lower α /2 tails of the sampling
   distribution (in this case for a central t with n − 1 df where
   n = n T = n C.

   
   


5. 
   

   Calculate td=− −{ˆ (μμTC d)}/[ /Sn], where S d is the standard
   deviation of the paired differences, and d

   
   

^

is the mean of the

paired differences.

6. Reject the null hypothesis if the test statistic t (computed in step
   5 above) falls in the rejection region for the test; otherwise, do
   not reject the null hypothesis.

The example of daily temperatures in Washington, DC, compared with
New York is next illustrate to dramatically depict the situations where
pairing works best. Although this is a weather example, similar improve-
ments can occur in clinical trials or epidemiology studies where pairing
is done by patient as in a cross - over trial or propensity score matching
in a case - control study. The paired data is given in Table 6.1.
The data are paired by date. We see that over the course of the year,
the average temperature varies periodically with the lowest tempera-
tures in the winter months and the highest in the summer. The date the
temperatures were taking is the 15th of the month for all months in the
year. Because New York and Washington are relatively close, they often
share the same weather system on a particular date. Note that from this
data, the highest mean temperature in Washington is 93 ° F occurring on
July 15 and the lowest mean temperature is 26 ° F on December 15. For
New York, the highest mean temperature is 89 ° F, also occurring on July
15, and the lowest mean temperature is 24 ° F on December 15. Over