__Sample size determination__:

Studies on Sample size determination according to Tongtong (2008) have concentrated on cross sectional studies. In his article he highlights two methods of determining the *required* sample size and power for repeated measures study design. He gives an example where a study in the medical field may want to determine the effectiveness of a new drug, one group is given drug A and group 2 is given drug B. the groups are then monitored for a period of time and data is collected from the same sample for that time duration. He refers to this as studies aimed at determining the “Change in mean response over time” (CIMROT) (Tongtong, 2008)

The article highlights two methods including the Diggle (1994) basic *sample size determination* and Fitzmaurice (2004) two stage method (Tongtong, 2008). Using these two methods the sample size for each group can be determined. Diggle (1994) method involves estimating a regression model where the response is a linear function of time and treatment **effect** is expressed as the differences in slope, if treatment effect does not exist then the difference between the slopes of the estimated models is zero.

The following is summary of Diggle (1994) method:

Assuming

There are two groups X and Y and each group is given a different drug

Measures are undertaken once a month for one year

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Sample Size Determination

Then the two models will be as follows:

Zx = β0 + βxT + e

Zy = β0 + βyT + e

Where T is time, z is the response or measure to determine the effect of the drug. Treatment effect will be determined by testing the hypothesis that H0; βx =βy

The next step is to rewrite the above equation in matrix form as follows:

Zi = X’ β(g) +e

Where

The Zi matrix is as follows:

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Sample Size Determination

For the first model β(g) = βx and for the second model β(g) = βy

The coefficients can be determined using the variance covariance matrix (Σ=σ2R) and for the models the estimated coefficients (β”

(g)

) are as follows:

β”(g) =(Σ X’ Σ-1Xi)-1 (Σ X’ Σ-1 Zi)

The assumption here is that the estimated coefficients are unbiased estimators of the true values of the slope and constant.

When testing the null hypothesis the statistics value is determined as follows:

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Sample Size Determination

T statistics = (βx – βy)/ (variance (βx – βy)) ½

Given that Type I error = α

And type II error rate is = β

Then the probability of rejecting the null hypothesis is as follows:

Power = 1- β= Pr( rejecting null hypothesis) = Pr(|T|) > Pr(T 1- α/2)

Therefore

Power =1- β= Pr( (βx – βy-D)/ (variance (βx – βy)) ½)> Pr(T 1- α/2) – Pr( (D)/ (variance (βx – βy)) ½)

Where

D is the difference (estimated difference between the slopes βx -βy)

Simplifying the above equation gives us:

-T 1- b = (T 1- α/2)- ((D)/ (variance (βx – βy)) ½)

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Sample Size Determination

[(T 1- α/2) + (T 1- b)] 2 = (D2/ (variance (βx – βy))

(D2/ (variance (βx – βy)) = [(Π (1- Π) M .N.S2)/ σ2 (1-P)]* D2

Making M the subject:

Total sample size = M = (T 1- α/2 + T 1- b )2 (1-P)S2

─────────────────

Π(1- Π)n S2D

Where

N= number of repeated observations

(T 1- α/2 + T 1- b) 2 = D2/ Var (βx -βy)

S2 =Standard deviation

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*Sample Size Determination*

D is the difference (estimated difference between the slopes βx -βy)

Π is the proportion of respondents in group X

P is the correlation between measures for a single individual.

This method can be applied to other fields of academics, in relation to job satisfaction this method can be used to determine the changes that occur to job satisfaction over time, for example when comparing two groups in different companies whereby the companies implements different strategies to increase job satisfaction. Over time the job satisfaction levels can be recorded to under take a repeated measures study. This method can be used to determine whether a significance difference in the strategies implemented is and **also** to determine the sample size required.

If the job satisfaction study was desired to be a repeated measures study then the required sample size, __assuming__ the following values are where the proportion Π to be 0.5, T 1- α/2 as 2.34,

T 1- b as

0.05, P = 0.6, S2 = 6.8 and D = 4 then the sample size will be determined as follows:

Total sample size = M = (T 1- α/2 + T 1- b) 2 (1-P) S2

─────────────────

Π(1- Π)n S2D

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Sample Size Determination

Total sample size = M = (0.5 + 0.05) 2 (1-0.6) 6.8

─────────────────

0.5(1- 0.5)(12). (0.7).(4)

M=10.44687

Therefore the appropriate sample size will be 10 respondents with 5 from each group and the study should be undertaken 12 times.

REFERENCE:

Tongtong Wu and Honghu Liu (2008). *Sample size determination*, Communications in Statistics, vol. 37, pages 1785 to 1798

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