Att7_SampleSize PowerCalc

Attachment 7

Anniston Community Health Survey: Follow-up Study and Dioxin Analyses

Sample Size and Power Estimation for Exposure Assessment

Part A. Sample size estimation for change in PCB congener value between Time 1 and Time 2

Study Design: One-time cross-sectional follow-up study seven years after baseline

Assumption 1: Metabolism and elimination of PCB congener follows an exponential decay model

Assumption 2: Additional individual exposure to PCB congener between Time 1 and Time 2 is minimal

Assumption 3: Change in PCB congener variable is lognormal

Null Hypothesis (H_O): PCB congener concentration follows an exponential decay model (Seegal et al., 2011) where half-life (HL) = 14 years (for PBCs 118, 153, 206 based on Knobeloch et al., 2009).

Alternative Hypothesis (H_A): PCB congener concentration follows and exponential decay model, where half-life (HL) = 20 years or longer.

, where

Let x₁, . . . ., x_n denote the data at Time 1. Then x₁ , . . . . , x_n are the data at Time 2.

The change in PCB level for individual, i, is given by:

Change = x_i – x_i = x_i (1– ).

Since we assume these variables are lognormal, we use a log-transformation to make the distribution normal.

g(x) ln (Change) = ln [x_i ] = ln (x_i) + ln (J),

where J is the constant, (1– ).

We can use a 2^nd order Taylor series around the mean, = to approximate Var (g(x)). [See Dudewicz & Mishra, p263, Theorem 5.5.18]

Var (g(x)) [g’ ( )]^{2 2},

where and ² are the mean and the variance of the individual data. It follows that

Var (g(x)) Var [ln (x_i) + ln (J)] .

The standard deviation of g(x) is approximated by .

Restated, the standard deviation of the natural log of the change in PCB level is approximately equal to the standard deviation of the individual data divided by their mean.

Before calculating a sample size, it is worth noting that the mean change in PCB level is equal to the change in the mean.

Mean Change in PCB Level = x₁ (1– ) + . . . . . . + x_n (1– )

n

= (1– )( x₁ + . . . . + x_n)

n

= (1– ) = Change in Mean PCB Level

Thus on the natural log scale,

Mean ln (Change in PCB Level) = ln ( ) + ln (J)

Example for representative low, moderately, and highly chlorinated PCB congeners using NCSS Power Analysis and Sample Size (PASS) 2008 Software, Kayesville, UT:

Test for One-Sample T-Test: H_O: HL = 14 years vs. H_A: HL ≥ 20 years

1. Under H_O, = 0.04951 and J = (1– ) = 0.2929

2. Under H_A, = 0.03466 and J = (1– ) = 0.2154

For PCB 118:

From Time 1, based on serum PCBs from n=765 ACHS participants, assume = 70, σ = 177, and Var [ln (Change)] = 177/70 = 2.5286.

H_O: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (70) + ln (0.2929)

= 3.0206

H_A: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (70) + ln (0.2154)

= 2.7132

A sample size of 420 achieves 80% power to detect a difference of 0.3 between the null hypothesis mean of 3.0 and the alternative hypothesis mean of 2.7 with an estimated standard deviation of 2.5 and = 0.05, using a one-sided one-sample t-test.

For PCB 153:

From Time 1, based on serum PCBs from n=765 ACHS participants, assume = 218, σ = 409, and Var [ln (Change)] = 409/218 = 1.8761.

H_O: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (218) + ln (0.2929)

= 5.3845 - 1.2279 = 4.1566

H_A: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (218) + ln (0.2154)

= 5.3845 - 1.5352 = 3.8492

A sample size of 232 achieves 80% power to detect a difference of 0.3 between the null hypothesis mean of 4.2 and the alternative hypothesis mean of 3.8 with an estimated standard deviation of 1.9 and with a significance level (alpha) of 0.05 using a one-sided one-sample t-test.

For PCB 206:

From Time 1, based on serum PCBs from n=764 ACHS participants, assume = 40, σ = 98, and Var [ln (Change)] = 98/40 = 2.4500.

H_O: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (40) + ln (0.2929)

= 3.68887945 - 1.22792403 = 2.4610

H_A: Mean ln (Change in PCB Level) = ln ( ) + ln (J) = ln (40) + ln (0.2154)

= 3.68887945 - 1.53525851 = 2.1536

A sample size of 395 achieves 80% power to detect a difference of 0.3 between the null hypothesis mean of 2.5 and the alternative hypothesis mean of 2.2 with an estimated standard deviation of 2.5 and with a significance level (alpha) of 0.05 using a one-sided one-sample t-test.

References:

Dudewicz, E.J., Mishra, S.N., 1988. Modern Mathematical Statistics. John Wiley & Sons, Inc., New York.

Knobeloch, L., Turyk, M., Imm, P., Schrank, C., Anderson, H., 2009. Temporal changes in PCB and DDE levels among a cohort of frequent and infrequent consumers of Great Lakes sportfish. Environ. Res. 109(1), 66-72.

Seegal, R.F., Fitzgerald, E.F., Hills, E.A., Wolff, M.S., Haase, R.F., Todd, A.C., Parsons P., Molho, E.S., Higgins, D.S., Factor, S.A., Marek, K.L., Seiby, J.P., Jennings, D.L., McCaffrey, R.J., 2011. Estimating the half-lives of PCB congeners in former capacitor workers measured over a 28-year interval. J. Exposure Sci. Environ. Epi. 21, 234–246.

Part B. Power estimation to detect difference in PCB levels between incident diabetes cases and non-diabetics

Method: Two-Sample T-Test Power Analysis

Estimated number of incident diabetes cases:

We estimate to enroll 365 persons without diabetes (out of 500 total). To estimate the total number of incident diabetes cases we used combined average rate of 9.5 /1,000 per year for 255 normoglycemic individuals and 50/1,000 for 110 pre-diabetics over the average of 7 years of follow up (See Section B.1 – Sample Size).

Supplemental Table 1. Estimated number of incident diabetes cases.

Year	No. Available	Rate per Year	No. Incident Cases		No. Available	Rate per Year	No. Incident Cases
	Normoglycemic				Pre-Diabetic
1.	255.0	0.0095	2.42		110.0	0.050	5.50
2.	252.6	0.0095	2.40		104.5	0.050	5.23
3.	250.2	0.0095	2.38		99.3	0.050	4.96
4.	247.8	0.0095	2.35		94.3	0.050	4.72
5.	245.4	0.0095	2.33		89.6	0.050	4.48
6.	243.1	0.0095	2.31		85.1	0.050	4.26
7.	240.8	0.0095	2.29		80.9	0.050	4.04
Total			16.48				33.18

The calculations suggest that we can assume to detect an estimated 16 cases of incident diabetes in normoglycemic individuals and about 33 cases in pre-diabetic individuals in 7 years of follow up for a total of 49 cases.

Report Definitions:

Power is the probability of rejecting a false null hypothesis. Power should be close to one.

N1 (non-diabetics) and N2 (incident diabetics) are the number of items sampled from each population.

Alpha is the probability of rejecting a true null hypothesis. It should be small.

Beta is the probability of accepting a false null hypothesis. It should be small.

Mean1 is the mean of populations 1 and 2 under the null hypothesis of equality.

Mean2 is the mean of population 2 under the alternative hypothesis. The mean of population 1 is unchanged.

S1 and S2 are the population standard deviations. They represent the variability in the populations.

Numeric Results for Two-Sample T-Test:

Null Hypothesis: Mean1=Mean2. Alternative Hypothesis: Mean1<Mean2.

Assumptions: The standard deviations were assumed to be unknown and unequal.

Power N1 N2 Ratio Alpha Mean1 Mean2 S1 S2

0.865 316 49 0.155 0.05000 6.1 6.6 1.3 1.2

Summary Statements:

Group sample sizes of 316 and 49 achieve 86.5% power to detect a difference of -0.5 between the null hypothesis that both group means are 6.1 and the alternative hypothesis that the mean of group 2 is 6.6 with estimated group standard deviations of 1.3 and 1.2 and with a significance level (alpha) of 0.0500 using a one-sided two-sample t-test.

References:

Machin, D., Campbell, M., Fayers, P., Pinol, A., 1997. Sample Size Tables for Clinical Studies, second edition. Malden, MA, Blackwell Science.

Zar, J.H., 1984. Biostatistical Analysis, second edition. Englewood Cliffs, NJ, Prentice-Hall.