Analysis of Treatment Means KNNL Chapter 17 Cell Means Model Sampling Distributions and Graphs Model Yij mi eij eij NID 0 s Fixed Effects Yij N mi s 2 1 mi Y i ni j 1 2 i 1 r j 1 ni independent s2 2 Yij E Y i mi s Y i ni s2 Y i mi Y i N mi independent t nT r s Y i ni ni s 2 Y i MSE ni Inference for Individual Treatment Means Y i mi s Y i t nT r where s Y i MSE ni 1 a 100 Confidence Interval for mi Y i t 1 a 2 nT r s Y i Test of H 0 mi c vs H A mi c Y i c Test Statistic t s Y i Reject H 0 if t t 1 a 2 nT r Note The t distribtion arises from 1 Y i mi s Y i N 0 1 2 SSE 2 c nT r s2 3 Y i SSE are independent Comparing Two Treatment Means Parameter D mi mi Difference between 2 Treatment Population Means Estimator D Y i Y i Difference between 2 Treatment Sample Means E D mi mi s s 2 2 1 1 D s 2 Y i s 2 Y i s 2 ni ni 1 1 D MSE ni ni 1 1 s D MSE ni ni D D t nT r 1 a 100 CI for D s D Test of H 0 D 0 mi mi vs H A D 0 D t 1 a 2 nT r s D mi mi Test Statistic t D s D Reject H 0 if t t 1 a 2 nT r Contrasts among Treatment Means Contrast A Linear Function of Treatment with their coefficients summing to 0 r L ci mi such that i 1 r r c i i 1 r L ci Y i 0 Note A difference between 2 means is contrast E L L ci mi i 1 s 2 i 1 s2 L c ni i 1 r L L t nT r 1 a 100 CI for L s L r r i 1 i 1 2 i r ci2 MSE s L c MSE n i 1 i 1 ni i 2 r 2 i L t 1 a 2 nT r s L Test of H 0 L ci mi 0 vs H A L ci mi 0 r Test Statistic t L s L c Y i i i 1 ci2 MSE i 1 ni r Reject H 0 if t t 1 a 2 nT r Note this method applies to any linear combination of means that is we do not need r c i i 1 0 Simultaneous Comparisons Confidence Coefficient 1 applies to only one estimate or comparison not several comparisons simultaneously Confidence Coefficient for a family of tests intervals will be smaller than confidence coefficient for individual tests intervals If we construct five independent confidence intervals each with confidence level 0 95 Pr All Correct 0 95 5 0 774 Confidence Coefficient 1 applies to only pre planned comparisons not those suggested by observed samples referred to as data snooping If we wait until after observing the data then decide to test whether most extreme means are different actual too high Tukey s Honest Significant Difference HSD I Background 1 Suppose Y1 Yr NID ms 2 and the range is w max Y1 Yr min Y1 Yr 2 s 2 is an estimate of s 2 based on n degrees of freedom 3 s 2 is independent of Y1 Yr w q r n is the studentized range with selected critical values in Table B 9 s Y Y w 5 P q r n q 1 a r n 1 a P i i q 1 a r n 1 a for all i i s s 4 Then Application to All Pairwise Comparisons Under Assumption of equal means and equal sample sizes s2 1a Y 1 Y r NID m n MSE s2 2a is an estimate of based on n nT r degrees of freedom n n 3a MSE independent of Y 1 Y r Y i Y i 4a P q 1 a r n 1 a for all i i MSE n Conclude any two population means are different if Y i Y i MSE n q 1 a r n Tukey s Honest Significant Difference HSD II Simultaneous Confidence Intervals all pairs of treatments D mi mi D Y i Y i 1 1 s D MSE ni ni Tukey s multiple confidence intervals with family level of 1 a D Ts D 1 where T q 1 a r nT r 2 Simultaneous tests of H 0 mi mi 0 vs H A mi mi 0 Test Statistic q 2D s D Reject H 0 if q q 1 a r nT r Scheffe s Method for Multiple Comparisons Very Conservative Method but can be applied to all possible contrasts among treatment means r L ci mi such that i 1 r L ci Y i i 1 r c i 0 i 1 ci2 s L MSE i 1 ni r Simultaneous 1 a 100 Confidence Intervals L Ss L S r 1 F 1 a r 1 nT r r r i 1 i 1 Testing H 0 L ci mi 0 vs H A L ci mi 0 2 Test Statistic F L r 1 s 2 L Reject H 0 if F F 1 a r 1 nT r Bonferroni s Method for Multiple Comparisons Can be used for any number g of pre planned comparisons contrasts linear combinations r L ci mi i 1 r L ci Y i i 1 ci2 s L MSE i 1 ni r Simultaneous 1 a 100 Confidence Intervals for g linear combinations of means L Bs L B t 1 a 2 g nT r r r i 1 i 1 Testing H 0 L ci mi 0 vs H A L ci mi 0 Test Statistic t L s L Reject H 0 if t t 1 a 2 g nT r