Sample Size Determination Text Section 3 7 pg

Sample Size Determination Text, Section 3 -7, pg. 101 • FAQ in designed experiments (what’s the number of replicates to run? ) • Answer depends on lots of things; including what type of experiment is being contemplated, how it will be conducted, resources, and desired sensitivity • Sensitivity refers to the difference in means that the experimenter wishes to detect • Generally, increasing the number of replications increases the sensitivity or it makes it easier to detect small differences in means

• Choice of sample size is closely related to the probability of type II error b. • Hypotheses H o: m 1 = m 2 H 1: m 1 m 2 • Type I error – reject H 0 when it is true (a) • Type II error – fail to reject H 0 when it is false (b) • Power = 1 – b = P(Reject Ho Ho is false) = P(Fo > Fa, a-1, N-a Ho is false) • The probability of type II error depends on the true difference in means d = m 1 - m 2

Operating Characteristic Curves for Fixed Effects/Equal Sample Sizes per Treatment Case § Operating characteristic curves plot b against a parameter F where § F is related to d. It depends on a (0. 01 and 0. 05) and degrees of freedom for numerator (a-1) and denominator (N-a). § Assumptions and trials are needed to use the curves

§ s 2 can be estimated through prior experience/ previous experiment/preliminary test/judgment, or assuming a range of likely values of s 2. • ith treatment effect where Assumed mi’s can be used for which we would like to reject the null hypothesis with high probability § Example (etch rate experiment) If the experimenter is interested in rejecting the null hypothesis with a probability of at least 0. 90 if the five treatment means are m 1 = 575 m 2 = 600 m 3 = 650 m 4 = 675

§ a = 0. 01 is planned § a = 4, N = an = 4 n, a – 1 = 3, N – a = 4(n-1) § is calculated using assumed mi’s § s is assumed no larger than 25 § § Find the right plot: § a – 1 = 3 (= v 1) determines the use of the upper plot on page 614 (Appendix Chart V) § Because a = 0. 01 the curves on the right side are used

Chart V. Operating characteristic curves for the fixed effects model analysis of variance (Page 614) the upper graph with v 1=3 should be used.

§ The objective is to find a b to see if the power is satisfied § It needs v 2 (or n) to determine the particular curve, and a value of F to determine b 2. 74 a(n-1) 8 0. 25 Power (1 - b) 0. 75 10. 0 3. 16 12 0. 04 0. 96 12. 5 3. 54 16 <0. 01 >0. 99 n 3 F 2 F 7. 5 4 5 § Therefore, at least n = 4 b

• It is often difficult to select a set of treatment means for choosing the sample size • A very common way to use these charts is to define a difference in two means D of interest, then the minimum value of F 2 is • Typically work in term of the ratio of D/s and try values of n until the desired power is achieved

Other Methods of Determining Sample Sizes Specifying a Standard Deviation Increase • As the difference between means increase, the standard deviation increases • Choose a percentage P for the increase in s of an observation beyond which the null hypothesis is rejected, equivalently

Specifying a Standard Deviation Increase (continued) • Rearrange it, • Therefore, F can be expressed as • Specify P, a, and the probability to reject the null hypothesis, then determine n.

Confidence Interval Estimation Method • Specify in advance how wide the confidence intervals should be by specifying the accuracy of the confidence interval • No OC curves are needed. Example: the etch rate experiment. • Need a and N (an) to determine ta/2, N-a, and s to estimate MSE • Specify the level of confidence (95%, or a =0. 05), difference in mean to be determined (30Å/min), and (prior) estimate s 2 (252 =625)

Confidence Interval Estimation Method (continued) • Procedure: compare the accuracy with an assumed n, with the specified accuracy (30Å/min) • When n = 5, ta/2, N-a =2. 120, • When n = 6, ta/2, N-a =2. 086, • When n = 7, ta/2, N-a =2. 064,

Dispersion Effects • Focus is location effects so far using ANOVA: factor level means and their differences • It needs constant variances. If not, using transformations to stabilize the variances. • Sometime the dispersion effects are of interest: whether the different factor levels affect variability • In such analysis, standard deviation, variance, or other measures of variability are used as response variables

An Example – Al Smelting Experiment • A reaction cell: Alumina and other ingredients (with a certain ratio) under electric resistance heating • Four different ratio control algorithms • Cell voltage is recorded (thousands of voltage measurements during each run) • A run consists of one ratio control algorithm • Average cell voltage (affecting cell temperature), and the standard deviation of cell voltage over a run (affecting overall cell efficiency) are response variables

Ratio Control Algorithm 1 2 3 4 5 6 1 4. 93(0. 05) 4. 86(0. 04) 4. 75(0. 05) 4. 95(0. 06) 4. 79(0. 03) 4. 88(0. 05) 2 4. 85(0. 04) 4. 91(0. 02) 4. 79(0. 03) 4. 85(0. 05) 4. 75(0. 03) 4. 85(0. 02) 3 4. 83(0. 09) 4. 88(0. 13) 4. 90(0. 11) 4. 75(0. 15) 4. 82(0. 08) 4. 90(0. 12) 4 4. 89(0. 03) 4. 77(0. 04) 4. 94(0. 05) 4. 86(0. 05) 4. 79(0. 03) 4. 76(0. 02) Observations • An ANOVA determines that the ratio control algorithm had no location effects (the ratio control algorithm does not change the average cell voltage) • A transformation is used to study the dispersion effects y = -ln(s) • A standard ANOVA can be done on y, the natural logarithm of standard deviation -> algorithm 3 produces different standard deviation than others