Ch11: Comparing 2 Samples11.1: INTRO:

This chapter deals with analyzing continuous measurements.

Later, some experimental design ideas will be introduced.

Chapter #13 will be devoted to qualitative data analysis.

11.2: Comparing Two Independent Samples

In medical study, one sample X of subjects may be assigned to a control (placebo) treatment and another sample Y to a particular (group) treatment.

This section deals with independent samples and later sections with dependent & paired samples.

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11.2.1: Methods based on Normal Distributions

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Null Hypothesis:

Test Statistic:

There 3 common alternative hypotheses. 2 of which are one-sided ( ) and one is two-sided ( ).

Revisit my handouts about CI and HT for references

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11.2.2 : Power calculationThe power of the 2-sample t-test depends on:

1. (real difference)– The larger , the greater the power

2. (level of significance)– The larger , the more powerful the test

3. (population standard deviation)– The smaller , the larger the power

4. n and m (sample sizes) – The larger n and m, the greater the power

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to test at level , with

test statistic based on ,

where are given.

The rejection region (RR) of such a test is:

The power of a test is the probability of rejecting the null hypothesis when it is false. That is,

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11.2.3: The Mann-Whitney Test(a nonparametric method)

Known as the Wilcoxon RST (Rank Sum Test).Assume that m + n experimental units are to be

assigned (at random) to a treatment group and a control group. In this specific context, n (remaining m) units are randomly chosen and assigned to the ctrl (to the trt).

We are interested in testing the null hypothesis that the treatment has NO EFFECT.

Then, if the null is true, then any difference in the outcomes under the 2 conditions is due to the randomization (i.e. solely by chance).

The Mann-Whitney Test: (cont’d)The MW-test statistic is calculated as follows:

1. Group all m + n observations together and Rank them in order of increasing size (no ties)

2. Calculate the sum of the ranks of those observations that came from the ctrl group.

3. Reject null if the sum is too small or too largeExample: ranks are bold and shown in parentheses

R = 3 + 4 = 7 (ctrl) and R = 1 + 2 = 3 (trt)

Treatment Control

1 (1) 6 (4)

3 (2) 4 (3)

The Mann-Whitney Test: (cont’d)Question: Does this discrepancy provide convincing

evidence of a systematic difference between trt & ctrl, or could it be just due by chance?

Answer: null hypothesis trt had no effect

Under the null, every assignment (total: 4!=24) of ranks to observations happens equally likely.

In particular, each of the assignments

of ranks to the ctrl group (shown below) is equally likely:

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The Mann-Whitney Test: (cont’d)The null distribution of R is the discrete r.v. R:

From this table, ; that is to say that this discrepancy would occur one time out of 6 by chance.

Similar computations can be carried out for any sample sizes m and n and can be even extended to testing:

Read page 404 (textbook).

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The Mann-Whitney Test: Another approach

Suppose that the X’s are sampled from F and the Y’s are sampled from G. The Mann-Whitney test can be derived from a different point of view than what was seen earlier.

We would like to estimate the probability that an observation from F is smaller than an independent observation from G which is as a measure of the treatment, where X and Y are independently distributed with distribution functions F and G.

An estimate of can be obtained by comparing all n values of X to all m values of Y and by calculating the proportion of the comparisons for which X is less than Y.

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11.3: Comparing Paired Samples

Paired Design vs Unpaired design:

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Pros & Cons Paired vs Independent Samples:

Here are 2 competiting sampling schemes:

Paired Samples: n pairs (2n measurements)

Independent Samples: 2n observations (m=n)

They both give the common form:

But, the SE estimates and the df for t are different:

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Pros & Cons Paired vs Independent Samples:

For a same SE estimate, a loss of DF (degrees of freedom) gives a larger value for the t-test.

(example: )A loss of DF for the t-test produces:

• C.I. Larger Confidence Intervals • H.T. Loss of Power to detect real differences in

the population means.Such loss of DF for Paired Samples is compensated

by a smaller variance Var(X—Y) of Paired Samples with respect to Independent Samples.

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11.3.2: Nonparametric Method for Paired Data: Sign Rank Test (SRT)

The Wilcoxon SRT is computed as follows:

1. Rank the absolute values of the differences (no ties) with

2. To get the signed ranks, just restore the signs of the to the ranks.

3. Calculate , the sum of those ranks that have positive (+) signs.

Example: Let be -2, 4, 3, 2, -1, 5

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11.4: Experimental Design

Some basic principles of DOE (Design of Experiment) are introduced here.

Experimental Design can be viewed as a sequence of linked studies under some conditions.

Read case studies 11.4.1 thru 11.4.8