How Can We Analyze Dependent Samples?

1
Section 9.4

• How Can We Analyze Dependent Samples?

2
Dependent Samples

• Each observation in one sample has a matched
  observation in the other sample
• The observations are called matched pairs

3
Example Matched Pairs Design for Cell Phones
and Driving Study

• The cell phone analysis presented earlier in this
  text used independent samples
• One group used cell phones
• A separate control group did not use cell phones

4
Example Matched Pairs Design for Cell Phones
and Driving Study

• An alternative design used the same subjects for
  both groups
• Reaction times are measured when subjects
  performed the driving task without using cell
  phones and then again while using cell phones

5
Example Matched Pairs Design for Cell Phones
and Driving Study

• Data

6
Example Matched Pairs Design for Cell Phones
and Driving Study

• Benefits of using dependent samples (matched
  pairs)
• Many sources of potential bias are controlled so
  we can make a more accurate comparison
• Using matched pairs keeps many other factors
  fixed that could affect the analysis
• Often this results in the benefit of smaller
  standard errors

7
Example Matched Pairs Design for Cell Phones
and Driving Study

• To Compare Means with Matched Pairs, Use Paired
  Differences
• For each matched pair, construct a difference
  score
• d (reaction time using cell phone) (reaction
  time without cell phone)
• Calculate the sample mean of these differences
  xd

8
For Dependent Samples (Matched Pairs)

• Mean of Differences
• Difference of Means

9
For Dependent Samples (Matched Pairs)

• The difference (x1 x2) between the means of the
  two samples equals the mean xd of the difference
  scores for the matched pairs
• The difference (µ1 µ2) between the population
  means is identical to the parameter µd that is
  the population mean of the difference scores

10
For Dependent Samples (Matched Pairs)

• Let n denote the number of observations in each
  sample
• This equals the number of difference scores
• The 95 CI for the population mean difference is

11
For Dependent Samples (Matched Pairs)

• To test the hypothesis H0 µ1 µ2 of equal
  means, we can conduct the single-sample test of
  H0 µd 0 with the difference scores
• The test statistic is

12
For Dependent Samples (Matched Pairs)

• These paired-difference inferences are special
  cases of single-sample inferences about a
  population mean so they make the same assumptions

13
Paired-difference Inferences

• Assumptions
• The sample of difference scores is a random
  sample from a population of such difference
  scores
• The difference scores have a population
  distribution that is approximately normal
• This is mainly important for small samples (less
  than about 30) and for one-sided inferences

14
Paired-difference Inferences

• Confidence intervals and two-sided tests are
  robust They work quite well even if the
  normality assumption is violated
• One-sided tests do not work well when the sample
  size is small and the distribution of differences
  is highly skewed

15
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• Boxplot of the 32 difference scores

16
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• The box plot shows skew to the right for the
  difference scores
• Two-sided inference is robust to violations of
  the assumption of normality
• The box plot does not show any severe outliers

17
Example Matched Pairs Analysis for Cell Phones
and Driving Study
18
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• Significance test
• H0 µd 0 (and hence equal population means for
  the two conditions)
• Ha µd ? 0
• Test statistic

19
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• The P-value displayed in the output is 0.000
• There is extremely strong evidence that the
  population mean reaction times are different

20
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• 95 CI for µd (µ1 - µ2)

21
Example Matched Pairs Analysis for Cell Phones
and Driving Study

• We infer that the population mean when using cell
  phones is between about 32 and 70 milliseconds
  higher than when not using cell phones
• The confidence interval is more informative than
  the significance test, since it predicts just how
  large the difference must be

22
Section 9.5

• How Can We Adjust for Effects of Other Variables?

23
A Practically Significant Difference

• When we find a practically significant difference
  between two groups, can we identify a reason for
  the difference?
• Warning An association may be due to a lurking
  variable not measured in the study

24
Example Is TV Watching Associated with
Aggressive Behavior?

• In a previous example, we saw that teenagers who
  watch more TV have a tendency later in life to
  commit more aggressive acts
• Could there be a lurking variable that influences
  this association?

25
Control Variable

• A control variable is a variable that is held
  constant in a multivariate analysis (more than
  two variables)

26
Can An Association Be Explained by a Third
Variable?

• Treat the third variable as a control variable
• Conduct the ordinary bivariate analysis while
  holding that control variable constant at fixed
  values
• Whatever association occurs cannot be due to
  effect of the control variable