Estimating Power and Sample Size for a One Sample t-test

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Logan Collins
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1 Biostatistics 130 Power & Sample Size 1 ORIGIN 0 Estimating Power and Sample Size for a One Sample t-test Pilot studies are often run in advance of collecting data for major statistical analyses. These studies are used to determine the POWER of an analysis - i.e., the ability of the analysis to satisfactorily lead to rejection of the Null Hypothesis and determining sufficient Sample Size to support sufficient power. Assumptions: - Observed values X 1, X, X 3,... X n are a random sample from ~N(, ). - Variance of the population is unknown. Estimating Sample Size for Desired Confidence Interval Width: ZAR READPRN ("c:/data/biostatistics/zarex7.3.txt" ) < Zar Example 7.3 ZAR X ZAR 1 n length( X) n 1 X bar mean( X) X bar s Var( X) s s Confidence Intervals (CI) are the primary means to estimate a population mean from sample data. The width of a CI can be controlled by sample size N for a given population variance. Zar 010 offers an interative procedure for estimating a minimum suitable sample size N. Note that N is the sample size we wish to estimate, not the sample size n of our pilot sample. Desired CI half width: d 0.5 < This is set as desired. Note that d is the CI width. Desired Type 1 Error level: 0.05 < Type 1 error Initial Guess of Sample Size: N 0 40 < Initial guess of sample sample size N needed. Iterative Calculation: s N 0 1 N 1 d N s N 1 1 N d N < This process is continued until the value of N i stabilizes. Then use the next largest interger value as an estimate of minimum sample size needed. ^ Note: input into the () function depends on whether the alternative two-sided test. For a one-sided test, use instead of /.

2 Biostatistics 130 Power & Sample Size #POWER & SAMPLE SIZE CALCULATIONS #ESTIMATING SAMPLE SIZE FOR CI #ZAR EXAMPLE 7.3 ZAR=read.table("c:/DATA/Biostascs/ZarEX7.3.txt") ZAR aach(zar) X=wtchg s=sqrt(var(x)) #sample variance d=0.5 N0=40 #ITERATE THE FOLLOWING UNTIL N IS STABILIZED: N1=(s^*((alpha/,N0 1))^)/d^ N1 N=(s^*((alpha/,N1 1))^)/d^ N N3=(s^*((alpha/,N 1))^)/d^ N3 N4=(s^*((alpha/,N3 1))^)/d^ N4 N5=(s^*((alpha/,N4 1))^)/d^ N5 Estimating Sample Size for a One Sample t-test This estimation differs from the one above in being specifically tied to a distance defined by the alternative H 0 and H 1 hypotheses of a one-sample t-test. ZAR READPRN ("c:/data/biostatistics/zarex7..txt" ) < Zar Example 7. X ZAR 1 n length( X) n X bar mean( X) X bar 0.65 ZAR s Var( X) s 1.53 s Hypotheses: 0 0 H 0 : = 0 H 1 : 0 < Let 0 = 0 < 0 is a specified value for < Two sided test Desired Precision : 1.0 < Set as desired for precision in estimating - 0. We want to reject H 0 if - 0 > Desired Type 1 & Error levels: < Type 1 error < Type error Initial Guess of Sample Size: N 0 0 < Initial guess of sample sample size N needed.

3 Biostatistics 130 Power & Sample Size 3 Iterative Calculation: s N 1 s N N 0 1 N 1 1 N 0 1 N 1 1 N N < This process is continued until the value of N i stabilizes. Then use the next largest interger value as an estimate of minimum sample size needed. #ESTIMATING SAMPLE SIZE FOR ONE SAMPLE T TEST #ZAR EXAMPLE 7. ZAR=read.table("c:/DATA/Biostascs/ZarEX7..txt") ZAR aach(zar) X=RATwtchg s=sqrt(var(x)) delta=1.0 beta=0.10 N0=0 Estimating Detectable Difference of a given Sample Size for a One Sample t-test For a given sample with size n, one can estimate = - 0 directly. Desired Sample Size: Desired Type 1 & Error levels: Calculation: N 5 s N ^ Note: input into the () function depends on whether the alternative two-sided test. For a one-sided test, use instead of /. < set for desired sample size N < Type 1 error < Type error < Note: sign of doesn't matter N 1 N 1 ^ Note: input into the () function depends on whether the alternative two-sided test. For a one-sided test, use instead of /. #ESTIMATING DETECTABLE DIFFERENCE GIVEN N #FOR ONE SAMPLE t TEST N=5 beta=0.10 delta=sqrt(s^/n)*((alpha/,n 1)+(beta,N 1)) delta #ITERATE THE FOLLOWING UNTIL N IS STABILIZED: N1=(s^/delta^)*((alpha/,N0 1)+(beta,N0 1))^ N1 N=(s^/delta^)*((alpha/,N1 1)+(beta,N1 1))^ N N3=(s^/delta^)*((alpha/,N 1)+(beta,N 1))^ N3 N4=(s^/delta^)*((alpha/,N3 1)+(beta,N3 1))^ N4 N5=(s^/delta^)*((alpha/,N4 1)+(beta,N4 1))^ N5

4 Biostatistics 130 Power & Sample Size 4 Estimating POWER of a One Sample t-test: POWER (1-) of a test is the probability of properly rejecting H 0 when it is false. It is the converse of Type error. We would like POWER to be as high as possible. Desired Sample Size: N 1 < set for desired sample size N Desired Precision : 1.0 < Set as desired for precision in estimating - 0. We want to reject H 0 if - 0 > Desired Type 1 Error level: 0.05 < Set Type 1 error Calculation: B s N N 1 B ^ Note: absolute value used here to allow use of standared () function whereas Zar uses a partial table that only has positive values. POWER pt( B N 1) POWER < Exact calculation using pt() function POWER N pnorm( B 0 1) POWER N ^ POWER = (1-) Note use of the probability functions. < Approximate calculation using pnorm() function assuming s=. #ESTIMATING POWER OF ONE SAMPLE t TEST N=1 delta=1.0 B=(delta/sqrt(s^/N)) abs((alpha/,n 1)) B POWER=pt(B,N 1) POWER POWERN=pnorm(B,0,1) POWERN

5 Biostatistics 130 Power & Sample Size 5

Tests for One Variance

Tests for One Variance Chapter 65 Introduction Occasionally, researchers are interested in the estimation of the variance (or standard deviation) rather than the mean. This module calculates the sample size and performs power