Lecture 5: Determining Sample Size - Purdue University

Statistics 514: Determining Sample Size

Fall 2021

Lecture 5: Determining Sample Size

Montgomery: Section 3.7 and 13.4

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Lecture 5 ? Page 1

Statistics 514: Determining Sample Size

Fall 2021

Choice of Sample Size: Fixed Effects ? Can determine the sample size for

? Overall F test

? Contrasts of interest

? For simplicity, typically assume ni's constant, i.e., n1 = n2 = ? ? ? = na = n

? Recall ? Type I error rate: = P(Reject H0|H0) ? Type II error rate: = P(Accept H0|H1) ? Power = P(Reject H0|H1) = 1-

? Need to know

? Test Statistics

? Distr. of test statistics under H0 = Reject Region (for given ) ? Distr. of test statistics under H1 = power = P(Reject Region | H1)

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Statistics 514: Determining Sample Size

Fall 2021

Determining Power for F Test

? = Pr(F0 > F,a-1,N-a|H0)

? = Pr(F0 < F,a-1,N-a|H1)

? Need to know distribution of F0 when H1 is true

? Can show F0 = MSTrt/MSE Fa-1,N-a() ? = n i2/2 (non-centrality parameter)

? Recall E(MSTrt)=2 + n i2/(a - 1) ? = {E(MSTrt) - E(MSE)} ? dfTrt/E(MSE)

? Need to specify {i}

(Note the zero-sum constraint:

a i=1

i

=

0)

? Power will vary for different choices

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Statistics 514: Determining Sample Size

Fall 2021

Power Calculation for F Test

? Given , a, and n, can determine F,a-1,N-a ? Given some value of , can use noncentral F to compute power

? In SAS, use function PROBF ? Power=1-PROBF(F,a-1,N-a,a - 1,N - a,) ? Montgomery: OCC given in Chart V ? Plots vs ? 2 = /a = n i2/(a2)

? Can use charts to determine power or sample size

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Statistics 514: Determining Sample Size

Methods to Determine or 2

1. Choose treatment means (? + i) ? Solve for {i} and compute 2 or

? Difficult to know what means to select

2. Take a minimum difference approach

? Suppose there exists a pair of (i, j) such that |i - j | D ? The minimum value: 2 = nD2/(2a2) (e.g.,

{i} = {-D/2, 0, . . . , 0, D/2}) ? Power of test is at least 1 -

3. Specify a standard deviation increase in percentage (P ) ? Under H1, variance of a randomly chosen yi is y2 = 2 + ? Randomly chosen i has mean 0 and variance i2/a ? P = 2 + i2/a - 1 ? 100 ? = an{(1 + .01P )2 - 1} ? 2 = n{(1 + .01P )2 - 1}

i2/a

Fall 2021

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