1. 1 Estimating and Testing  2 0 (n-1)s 2 /  2 has a  2 distribution with n-1 degrees of freedom Like other parameters, can create CIs and hypothesis tests since we know the distribution

2. 2 Estimating and Testing  1 2 /  2 2 0  1 2 /  2 2 has an F distribution with n 1 -1 and n 2 -1 d.f. Like other parameters, can create CIs and hypothesis tests since we know the distribution

3. 3 One-Way ANOVA Example As a training specialist, you want to determine whether three different training methods are equally effective so you want to compare the mean time to complete a task after individuals are trained with the three different methods. (Random variable: completion time) One-Way ANOVA hypothesis test H 0 :  1 =  2 =  3 =... =  k H A : The means are not all equal (at least one pair differs from each other) Three Requirements k independent random samples Random variables are normally distributed Equal population variances for the random variables

4. 4 One-Way ANOVA Data Set Sample One Sample Two...Sample k Data Values... Meanx1x1 x2x2 xkxk Variances12s12 s22s22...sk2sk2 Sample Sizen1n1 n2n2 nknk x 1,1 x 1,2 x 1,3 … x 2,1 x 2,2 x 2,3 … x k,1 x k,2 x k,3 … ___

5. 5 One-Way ANOVA One-Way ANOVA hypothesis test H 0 :  1 =  2 =  3 =... =  k H A : The means are not all equal (at least one pair differs from each other) Test Statistic F = BSV / WSV with k-1, n T -k d.f. Decision Rule Reject H 0 if F > F  Intuition If BSV is “large” then H 0 is unlikely to be true Note: Always one-tailed and >

6. 6 F Distribution 0 ? P(F > ) = 0.05 P(F < ) = 0.95 9 and 10 d.f

7. 7 Select F Distribution 5% Critical Values Numerator Degrees of Freedom 12456789… 1161199225230234237239241 218.519.019.219.3 19.4 310.19.559.129.018.948.898.858.81 85.324.463.843.693.583.503.443.39 104.964.103.483.333.223.143.073.02 114.843.983.363.203.093.012.952.90 124.753.893.263.113.002.912.852.80 184.413.552.932.772.662.582.512.46 1003.943.092.462.312.192.102.031.97 10003.853.002.382.222.112.021.951.89 … Denominator Degrees of Freedom

8. 8 One-Way ANOVA: Example H 0 :  1 =  2 =  3 H A : The means are not all equal Test Statistic F = BSV / WSV with k- 1,n T -k d.f. Decision Rule Reject H 0 if F > F  Intuition If BSV is “large” then H 0 is unlikely to be true Sample One Sample Two Sample Three 101620 101420 111620 101420 9 x s2s2 n _ n T =, x = =

9. 9 One-Way ANOVA: Another Example H 0 :  1 =  2 =  3 H A : The means are not all equal Test Statistic F = BSV / WSV with k- 1,n T -k d.f. Decision Rule Reject H 0 if F > F  Intuition If BSV is “large” then H 0 is unlikely to be true Sample One Sample Two Sample Three 1195 123133 204 10612 730 x s2s2 n _ n T =, x = =

10. 10 Data Analysis: ANOVA-Single Factor GroupsCountSumAverageVariance Column 1550100.5 Column 2460151.333 Column 35100200 ANOVA Source of Variation SSdfMSFP- value F crit Between Groups 2502125229.1670.0003.98 Within Groups 6110.545 Total25613

11. 11 Two-Way ANOVA Example Suppose 10 individuals are asked to judge the taste quality of three beers: Budweiser, Bass Ale, and London Pride. Based on some complicated rating system, each individual assigns a numerical score to each beer (all 10 individuals taste each of the three beers). Note that we now have two factors: raters and beers Questions Is the average taste rating equal for each beer? Is the average taste rating equal for each rater? Can do an ANOVA hypothesis test for each question: H 0 :  1 =  2 =  3 =... =  k H A : The means are not all equal (at least one pair differs from each other)

12. 12 Two-Way ANOVA: Example Suppose 10 individuals are asked to judge the taste quality of three beers: Budweiser, Bass Ale, and London Pride. Based on some complicated rating system, each individual assigns a numerical score to each beer (all 10 individuals taste each of the three beers). Once the data are collected, you estimate the following ANOVA table: Sourcedf Sum of Squares Mean Square F Beer25.982.992.39 Rater9112.4913.6110.89 Error1822.501.25

13. 13 Another Example Treatments ABC Blocks 11098 21265 3181514 42018 5878 These data values were obtained from a randomized block design experiment. First, Pretend that these data were collected using a completely randomized design. Can we conclude at the 5% significance level that there are differences in the treatment means? How does your answer change if you account for the randomized block design?

14. 14 Pretend One-Way GroupsCountSumAverageVariance Treatment A56813.6026.80 Treatment B55511.0027.50 Treatment C55310.6027.80 ANOVA Source of Variation SSdfMSFP- value F crit Between Groups 26.53320.6273.86 Within Groups 328.40012 Total354.93314

15. 15 But is Actually Two-Way ANOVA: Two-Factor Without Replication Source of Variation SSdfMSFP- value F crit Rows312.267478.06738.7110.0003.84 Columns26.533213.2676.5790.0204.46 Error16.13382.017 Total354.93314

16. 16 One-Way ANOVA via Regression Sample One Sample Two Sample Three 101620 101420 111620 101420 9 S1S2S3Treatment 10010 100 10011 10010 1009 01016 01014 01016 01014 00120 001 …… “Stack” data using dummy variables

17. 17 One-Way ANOVA via Regression ANOVA (from ANOVA output) Source of Variation SSdfMSFP- value F crit Between2502125229.1670.0003.98 Within6110.545 Total25613 ANOVA (from Regression output) dfSSMS FSignificance Regression2250125229.1670.0000 Residual1160.545 Total13256

18. 18 The Regression ANOVA Table Regression Statistics Multiple R0.988 R Squared0.977 Adj. R Squared0.972 Standard Error0.739 Obs.14 ANOVA dfSSMSFSignificance Regression2250125229.1670.0000 Residual1160.545 Total13256 Coeff.Std. Errort statp valueLower 95%Upper 95% Intercept100.33030.2770.00009.27310.727 S250.49510.0920.00003.9106.090 S3100.46721.4090.00008.97211.028 F test for: Ho:  1 =  2 = … =  k = 0 (excluding the intercept) H A : at least one  i  0

19. 19 Regression and Two-Way ANOVA Treatments ABC 11098 21265 3181514 42018 5878 Blocks “Stack” data using dummy variables ABCB2B3B4B5Value 100000010 100100012 100010018 100001020 10000018 01000009 01010006 010010015 010001018 01100017 00100008 ……

20. 20 Regression and Two-Way ANOVA Source | SS df MS Number of obs = 15 ----------+---------------------- F( 6, 8) = 28.00 Model | 338.800 6 56.467 Prob > F = 0.0001 Residual | 16.133 8 2.017 R-squared = 0.9545 -------------+------------------- Adj R-squared = 0.9205 Total | 354.933 14 25.352 Root MSE = 1.4201 ------------------------------------------------------------- treatment | Coef. Std. Err. t P>|t| [95% Conf. Int] ----------+-------------------------------------------------- b | -2.600.898 -2.89 0.020 -4.671 -.529 c | -3.000.898 -3.34 0.010 -5.071 -.929 b2 | -1.333 1.160 -1.15 0.283 -4.007 1.340 b3 | 6.667 1.160 5.75 0.000 3.993 9.340 b4 | 9.667 1.160 8.34 0.000 6.993 12.340 b5 | -1.333 1.160 -1.15 0.283 -4.007 1.340 _cons | 10.867.970 11.20 0.000 8.630 13.104 ------------------------------------------------------------- Need Partial F test (later in course)