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Dependent Samples

With two dependent samples, the observations in the two samples are somehow “linked” between the samples by design (e.g., same unit observed twice, twins). A more technical definition is that the distribution of an observation in one sample would depend on the value of an observation in the other sample. Matched-pairs designs create dependent samples.

Example: Consider the following data from a study of the effect of in-synchrony versus out-of-sychrony speech on infant attention.1
Synchrony
Infant Out In Diff
DC 50.4 20.3 30.1
MK 87.0 17.0 70.0
BH 25.1 6.5 18.6
JM 28.5 25.0 3.5
SB 26.9 5.4 21.5
MM 36.6 29.2 7.4
RH 1.0 2.9 -1.9
DJ 43.8 6.6 37.2
JD 44.2 15.8 28.4
ZC 10.4 8.3 2.1
CW 29.9 34.0 -4.1
AF 27.7 8.0 19.7

The observations are of the percent time attending to the person speaking. The average difference is about 19.4% and the standard deviation of the differences is about 20.8%.

Independent Samples

With two independent samples, the observations in the two samples are not dependent.

Example: Had the study of the effect of in-synchrony versus out-of-synchrony speech used a randomized design in which each infant was assigned to either the in-synchrony or out-of-sychrony condition, it would produce independent samples that might look like this.
Infant Synchrony Attention
DC In 20.3
MK Out 87.0
BH Out 25.1
JM In 25.0
SB In 5.4
MM Out 36.6
RH In 2.9
DJ In 6.6
JD Out 44.2
ZC Out 10.4
CW In 34.0
AF Out 27.7
The mean, standard deviation, and sample sizes for the two sample are given below.
Synchrony \(\bar{x}\) \(s\) \(n\)
In 15.7 12.6 6
Out 38.5 26.4 6
Example: A large-scale randomized experiment investigated the effect of regular aspirin use on myocardial infarctions (i.e., heart attacks).2
Subject Treatment Heart Attack?
1 aspirin no
2 aspirin no
3 control yes
4 aspirin no
5 aspirin no
6 control yes
7 control yes
8 aspirin no
9 control no
10 aspirin yes
\(\vdots\) \(\vdots\) \(\vdots\)
22071 aspirin no
The data can be summarized as follows.
Heart Attack?
Group yes no Total
aspirin 104 10933 11037
control 189 10845 11034

Sampling Distribution of \(\hat{p}_1 - \hat{p}_2\)

What do we know about the sampling distribution of \(\hat{p}_1 - \hat{p}_2\)?

  1. The mean of the sampling distribution is \(p_1 - p_2\).

  2. The standard deviation (i.e., standard error) is \[ \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}, \] assuming independent samples. Why? Because the variance of the sum or difference of two indepedent random variables is equal to the sum of their variances.

  3. The shape of the sampling distribution is approximately that of a normal probability distribution by an application of the central limit theorem.

Confidence Interval for \(p_1-p_2\)

A general “recipe” for a confidence interval is \[ \text{point estimate} \pm \text{margin of error}, \] where \[ \text{margin of error} = \text{multiplier} \times \text{standard error}. \] How then might we compute the point estimate, margin of error, and confidence interval for \(p_1-p_2\), and how might we apply that to the results of the aspirin study?
Heart Attack?
Group yes no Total
aspirin 104 10933 11037
control 189 10845 11034

Statistical Test Concerning \(p_1-p_2\)

A general “recipe” for a test statistic is \[ \text{test statistic} = \frac{\text{point estimate} - \text{hypothesized mean}}{\text{standard error}}. \] How might we conduct a statistical test concerning \(p_1-p_2\), and how might we apply that to the results of the aspirin study?
Heart Attack?
Group yes no Total
aspirin 104 10933 11037
control 189 10845 11034

Example: A study published in The New England Journal of Medicine reported the results of a randomized experiment with 128 children and adolescents to investigate the effectiveness of the drug fluvoxamine in the treatment of anxiety disorders in young people.3 The study found that 48 out of 63 subjects that were given the drug showed a reduction in anxiety, in comparison to only 19 out of 65 subjects that were not given the drug. What can we infer about the effect of the fluvoxamine on anxiety reduction?

  1. Dodd, B. (1979). Lip reading in infants: Attention to speech presented in- and out-of-synchrony. Cognitive Psychology, 11, 478–484.↩︎

  2. Steering Committee of the Physicians’ Health Study Research Group. (1989). Final report on the aspirin component of the ongoing Physicians’ Health Study. New England Journal of Medicine, 321, 129–135.↩︎

  3. Walkup, J. T. et al. (2001). Fluvoxamine for the treatment of anxiety disorders in children and adolescents. The New England Journal of Medicine, 344, 1279–1285.↩︎