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STATISTICAL INFERENCE PART IX

STATISTICAL INFERENCE PART IX. HYPOTHESIS TESTING - APPLICATIONS – MORE THAN TWO POPULATION. INFERENCES ABOUT POPULATION MEANS. Example: H o :  1 =  2 =  3 where  1 = population mean for group 1  2 = population mean for group 2  3 = population mean for group 3

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STATISTICAL INFERENCE PART IX

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  1. STATISTICAL INFERENCEPART IX HYPOTHESIS TESTING - APPLICATIONS – MORE THAN TWO POPULATION

  2. INFERENCES ABOUT POPULATION MEANS • Example: • Ho: 1 = 2 = 3 where • 1 = population mean for group 1 • 2 = population mean for group 2 • 3 = population mean for group 3 • H1: Not all are equal.

  3. Assumptions • Each of the populations are normally distributed (or large enough sample sizes to use CLT) with equal variances • Populations are independent • Cases within each sample are independent

  4. INFERENCES ABOUT POPULATION MEANS - ANOVA Difference in means large relative to overall variability Difference in means small relative to overall variability  F tends to be small  F tends to be large Larger F-values typically yield more significant results. How large is large enough? We will compare with the tabulated value.

  5. INFERENCES ABOUT POPULATION MEANS • If F test shows that there are significant differences between means, then, apply paired t-tests to see which one(s) are different. • Apply multiple testing correction to control for Type I error

  6. Example • Kenton Food Company wants to test 4 different package designs for a new product. Designs are introduced in 20 randomly selected markets. These markets are similar to each other in terms of location and sales records. Due to a fire incidence, one of these markets are removed from the study, leading to an unbalanced study design. Example is taken from: Neter, J., Kutner, M.H., Nachtsheim, C.J., & Wasserman, W., (1996) Applied Linear Statistical Models, 4th edition, Irwin.

  7. Example Is there a difference among designs in terms of their average sales?

  8. Example > va1=read.table("VAT1.txt",header=T) > head(va1,3) Case Design Market Sales 1 1 1 1 11 2 2 1 2 17 3 3 1 3 16 > aov1 = aov(Sales ~ Design,data=va1) > summary(aov1) Df Sum Sq Mean Sq F value Pr(>F) Design 1 483.08 483.08 31.186 3.289e-05 *** Residuals 17 263.34 15.49 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Degrees of freedoms are wrong! Since there are 4 different designs, d.f. should be 3.

  9. Example > class(va1[,2]) [1] "integer" > va1[,2]=as.factor(va1[,2]) > aov1 = aov(Sales ~ Design,data=va1) > summary(aov1) Df Sum Sq Mean Sq F value Pr(>F) Design 3 588.22 196.074 18.591 2.585e-05 *** Residuals 15158.20 10.547 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 # or, alternatively: > aov1 = aov(Sales ~factor(Design),data=va1) 4 designs have different mean sales. But, which one(s) are different?

  10. Example > library(multcomp) > c1=glht(aov1, linfct = mcp(Design = "Tukey")) > summary(c1) Simultaneous Tests for General Linear Hypotheses Multiple Comparisons of Means: Tukey Contrasts Fit: aov(formula = Sales ~ Design, data = va1)  Linear Hypotheses: Estimate Std. Error t value Pr(>|t|) 2 - 1 == 0 -1.200 2.054 -0.584 0.9352 3 - 1 == 0 4.900 2.179 2.249 0.1545 4 - 1 == 0 12.600 2.054 6.135 <0.001 *** 3 - 2 == 0 6.100 2.179 2.800 0.0584 . 4 - 2 == 0 13.800 2.054 6.719 <0.001 *** 4 - 3 == 0 7.700 2.179 3.534 0.0141 * Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Adjusted p values reported -- single-step method) 4th design has higher average sales than all other designs. 3rd design is slightly significantly better than 2nd design.

  11. Example # or, alternatively > TukeyHSD(aov1, "Tasarim", conf.level=0.9) • There are many functions in R available for multiple testing correction. For instance, you can look into “p.adjust” function in stats library for other types of corrections (e.g. Bonferroni). Supply raw p-values  obtain adjusted p-values. • Different ANOVA types (e.g. 2-factor, repeated,…) in R; reference: Ilk, O. (2011) R YaziliminaGiris, ODTU, Chp. 7

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