Background
� Paired Samples � Illustrative Data Set (OATBRAN.SAV)
Descriptive Statistics
� Creating DELTA � Notation � Summary Statistics
Inferential Statistics
� Estimation � Hypothesis Testing � Assumptions � Power of the t Test
This chapter considers the analysis of a continuous outcome based on paired samples. An example of this type of sample is the "pre-test/post-test" sample in which a single group is measured before and after an intervention. However, paired samples can be obtained by other means as well. For example, we could perform a cross-over trial in which a patient is tried on one treatment followed by another, or we can match on important intrinsic or extrinsic characteristic, such as age or sex. Regardless of how pairing is achieved, natural pairs must be considered throughout subsequent analyses.
Let us use the data set OATBRAN.SAV to illustrate principals in this chapter. These data were collected to investigate whether oat bran helps lower serum cholesterol levels in hypercholesterolemic patients (Pagano & Gauvreau, 1993, p 252-253). Fourteen individuals were randomly assigned a diet that included either oat bran or corn flakes. After two weeks on their initial diet, low-density lipoprotein levels (LDL, in mg/dl) were measured. Each subject was subsequently switched to the alternate diet. After two-weeks on this second diet, LDL was once again recorded. Data are:
ID CORNFLK OATBRAN
------- ------- -------
1 4.61 3.84
2 6.42 5.57
3 5.40 5.85
4 4.54 4.80
5 3.98 3.68
6 3.82 2.96
7 5.01 4.41
8 4.34 3.72
9 3.80 3.49
10 4.56 3.84
11 5.35 5.26
12 3.89 3.73
13 2.25 1.84
14 4.24 4.14
We wish to describe the effect of diet on LDL levels.
Suggestion: Enter these data set into an SPSS file in preparation for analysis.
The first step in this type of analysis is to create a difference variable called DELTA. This can be done by simple subtraction. Let X1 represent the first measurement, X2 represent the second measurement, and DELTA represent the difference between these measurements. Thereby,
DELTA = X1 - X2
The illustrative data set with DELTA values listed is:
ID CORNFLK OATBRAN DELTA
1 4.61 3.84 .77
2 6.42 5.57 .85
3 5.40 5.85 -.45
4 4.54 4.80 -.26
5 3.98 3.68 .30
6 3.82 2.96 .86
7 5.01 4.41 .60
8 4.34 3.72 .62
9 3.80 3.49 .31
10 4.56 3.84 .72
11 5.35 5.26 .09
12 3.89 3.73 .16
13 2.25 1.84 .41
14 4.24 4.14 .10
The DELTA variable can now be explored with a stem-and-leaf plot:
|-0|24
| 0|011334
| 0|667788
Change in LDL level (mg/dl x 0.1)
Notice that all but 2 of the 14 observations in this data set show lower LDL levels when on oatbran and that the center of the distribution is to the right of 0.
SPSS: To create difference variable DELTA in SPSS, click on Transform | Compute, and then type DELTA in the Target Variable dialogue box and CORNFLK - OATBRAN in the Numeric Expression dialogue box. This subtracts each OATBRAN value from its associated CORNFLK value, and puts this information difference into the DELTA variable. Values for this new DELTA variable are shown below:
We wish to describe each of the paired samples separately, and the difference between them. Subscripts are used to denote the sample being described. For example, x1 will be used to denote the mean of sample 1, s1 will denote the standard deviation of sample 1, x2 will be used to denote the mean of sample 2, and so on. Statistics associated with the paired difference will be denoted with a subscript of d (e.g., x, sd, etc.). The sample sizes, n1 = n2 = n, represents the number of paired observations.
The statistics for the separate samples are calculated in the usual way, either by hand or with the assistance of a computer program.
For the illustrative data set, the CORNFLK variable has a mean (x1) of 4.444 and standard deviation (s1) of 0.969. The OATBRAN variable has a mean (x2) of 4.081 and standard deviation (s2) of 1.057. LDL levels are lower on average with OATBRAN than with CORNFLK. The difference in LDL levels is inherent in the DELTA variable. The mean of DELTA (xd) is 0.363 and standard deviation (sd) is 0.4060. The total number of pairs (n) = 14.
SPSS: SPSS provides the mean and standard deviation of the difference as a matter of course when you use the Statistics | Compare Means | Paired Sample T command.
Inferential statistics for paired samples is just the same as inferential statistics for a single sample, however all results are now directed toward difference variable DELTA. In practice, we have converted to variables (the "before" and "after" values) to a single variable. All statistical operations of inference are now directed toward this new (user-created) variable. Accordingly, the sample mean difference xd is the point estimator of population mean (expected) difference �d (0.3629 mg/dl, for the illustrative data set).
A 95% confidence interval for �d is calculated with the formula:
xd � (tn - 1,.975)(se)
where tn - 1,.975 represents the 97.5th percentile of a t distribution with n - 1 degrees of freedom and se represents the estimated standard error of DELTA (se = sd / sqrt(n)). For our illustrative example, xd = 0.3629; sd = 0.4060, se = 0.4060 / sqrt(14) = 0.1085, n = 14, and t13, .975 = 2.16. Therefore, a 95% confidence interval for �d = 0.3629 � (2.16)(0.1085) = 0.3629 � 0.2344 = (0.1285, 0.5973). We can now say with 95% confidence that the population mean decrease in LDL levels lies between 0.1285 mg/dl and 0.5973 mg/dl when on oatbran as compared with corn flakes.
SPSS: Statistics computes the confidence interval with the Statistics | Compare Means | Paired Samples T Test command, as described earlier.
Once again, we direct our inferences toward difference variable DELTA. The most intuitive question to ask in addressing these data is whether the observed change is significantly different from 0. The most common two-sided null and alternative hypotheses are therefore:
H0: �d = 0
H1: �d not = 0
where �d represents the population mean (expected) difference.
The a level of the test should be set before calculating test statistics. (Let a = 0.05, unless otherwise specified.)
This test is called the paired t test. The test statistic is:
t-stat = xd / (se)
For the illustrative example, t-stat = 0.3629 / 0.1085 = 3.344. This statistic has n - 1 degrees of freedom (illustrative example, df = 14 - 1 = 13). Under the null hypothesis, the p value is the area under the tn-1 distribution beyond the absolute value of the t statistic. For the illustrative data set p (two-sided) = 0.005. Therefore, data are significant.
SPSS: The paired t test is computed by clicking on Statistics | Compare Means | Paired Samples T Test.
The above inferential statistics assume data are representative, measurement error is minimal, and the sampling distribution of the mean differences is normal. Because of the central limit theorem, we do not usually concern ourselves with the normality assumption, as long as data are mound-shaped or the sample is "large" (say, n >= 30).
If H0 is false and we fail to reject it, we commit a type II error. The probability of avoiding a type II error is the power of the test. The power of a paired t test, depends on:
Given these factors, the power of a t test performed at a = 0.05 (two-sided) is approximately equal to the area on a standard normal curve to the left of:
-1.96 + {|d| * sqrt(n) / sd}
(Rosner, 2000, p. 235).
Illustrative example: Suppose you want to determine the power of a paired t test in which you are expecting a mean difference of 2, a (two-sided) = 0.05, n = 30, and the standard deviation of the difference = 6. The power of the test is the area on a standard normal curve to the right of -1.96 + {|2| * sqrt(30) / 6}} = -0.13. The area to the left of -0.13 on a standard normal curve = 0.45. We generally want the power of a test to exceed 80%. Therefore, the power of this test was inadequate.
A Web-based power located at the URL http://www.health.ucalgary.ca/~rollin/stats/ssize/n1.html can be used to compute this probability. This Web calculator listed above (which gives a power of 0.423). By either estimate, the power of the test is inadequate.