C2M2_peer_reviewed

hardness %>% ggplot(aes(y = y, x = tip)) + geom_line(aes(group = coupon, color = coupon)) + labs(x = "Tip Type" , color = "Coupon Type" , y = "Hardness" , title = "Line Plots of Tip type Vs. Hardness, group by Coupon Type" ) hardness %>% ggplot(aes(x = tip, y = y)) + geom_point(aes(color = coupon)) + facet_wrap( ~ coupon) + labs(x = "Tip Type" , y = "Hardness" , color = "Coupon Type" , title = "Scatterplot for Tip type Vs. Hardness, Faceted by Coupon Type" ) `geom_smooth()` using formula 'y ~ x' 3

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(Intercept) 9.35000 0.06236 149.934 < 2e-16 *** tip2 0.02500 0.06667 0.375 0.716345 tip3 -0.12500 0.06667 -1.875 0.093550 . tip4 0.30000 0.06667 4.500 0.001489 ** coupon2 0.02500 0.06667 0.375 0.716345 coupon3 0.32500 0.06667 4.875 0.000877 *** coupon4 0.55000 0.06667 8.250 1.73e-05 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.09428 on 9 degrees of freedom Multiple R-squared: 0.938,Adjusted R-squared: 0.8966 F-statistic: 22.69 on 6 and 9 DF, p-value: 5.933e-05 The second model I fitted, which includes predictors for “tip” and “coupon” without an interaction term, offers a more straightforward interpretation of the data. The analysis indicates several key insights into how these variables impact hardness. Firstly, the intercept suggests that the expected hardness is around 9.35 when both “tip” and “coupon” are at their baseline levels. When examining the coefficients for “tip,” only the level “tip4” shows a statistically significant effect on hardness, with an increase of 0.3 units compared to the baseline. On the other hand, for “coupon,” all levels except “coupon2” exhibit significant effects on hardness, with “coupon4” having the most substantial impact, leading to a 0.55 unit increase. These findings are supported by the model’s high adjusted R-squared value of 0.8966, indicating that approximately 89.66% of the variability in hardness can be explained by the predictors. 2.0.3 1. (c) Contrasts Let’s take a look at the use of contrasts. Recall that a contrast takes the form t ∑ i =1 c i μ i = 0 , where c = ( c 1 , ..., c t ) is a constant vector and μ = ( μ 1 , ..., μ t ) is a parameter vector (e.g., μ 1 is the mean of the i th group). We can note that c = (1 , - 1 , 0 , 0) corresponds to the null hypothesis H 0 : μ 2 - μ 1 = 0 , where μ 1 is the mean associated with tip1 and μ 2 is the mean associated with tip2. The code below tests this hypothesis. Repeat this test for the hypothesis H 0 : μ 4 - μ 3 = 0 . Interpret the results. What are your conclusions? [17]: library(multcomp) lmod = lm(y ~ tip + coupon, data = hardness) 9

4 Problem 3: Post-Hoc Tests There’s so many different post-hoc tests! Let’s try to understand them better. Answer the following questsions in the markdown cell: • Why are there multiple post-hoc tests? • When would we choose to use Tukey’s Method over the Bonferroni correction, and vice versa? • Do some outside research on other post-hoc tests. Explain what the method is and when it would be used. 1. Post-hoc tests serve to address the familywise-error problem in statistical analyses, with various methods available for correction. There is no consensus on a universally superior approach, leading to the selection of different tests based on the research context and priorities. 2. The Bonferroni method offers certain advantages and potential drawbacks compared to the Tukey method. It tends to be slightly conservative, providing alpha values slightly below the desired cutoff, which can be beneficial or limiting depending on the situation and the number of tests conducted. As the number of tests increases, the Bonferroni correction’s accuracy diminishes, impacting its power compared to Tukey, especially for the same number of tests. Bonferroni allows for testing a limited number of simultaneous hypotheses, whereas Tukey automatically conducts pairwise comparisons, which can be resource-intensive for variables with numerous factor levels. Tukey is particularly effective for factors with few levels and allows precise alpha specification using the Studentized Range Distribution. 3. Exploring post-hoc tests further, Tukey’s HSD test is a post-hoc test commonly used in ANOVA to identify which specific groups or treatments differ significantly from each other. It compares all possible pairs of means and determines whether the difference between any two means is statistically significant. Tukey’s HSD is robust and widely used, especially when the sample sizes are equal. It helps researchers gain a deeper understanding of the differences among groups identified as significant in the initial ANOVA analysis, providing valuable insights into the relationships between variables in the study. [ ]: 12