2. Refer to the “Dental Gold” data set posted on our Courselink website. This data set is a subset of a larger experiment; we are using data from only one of the two alloys originally included in the experiment. Our data set will be used to analyze the effects of three factors (Method, Temperature, and Dentist) on the hardness of dental fillings. Each of five dentists prepared the gold alloy by each of the three methods of condensation and at each of the three temperatures used in the molting process (temperatures in EF). The dentist prepared a filling in an artificial substrate from each alloy made, and the hardness was measured via the Vickers Hardness Test.

Note the resulting data set contains 45 observations, one for each treatment combination.

Read the data set into R. The levels of each of the three factors are numerically coded, so they will have to be declared as factors prior to subsequent analysis. Prior to declaring Temperature as a factor, save a copy of this variable as a numeric variable (e.g, Temp.Numeric <-Temperature).

(a) Refer to the handout “Model Equations for Some Basic Experimental Designs”. Try implementing an analysis based on equation (6): the aov() model statement would be Hardness ~ Method*Temperature*Dentist. You should encounter a problem with the analysis. What problem do you encounter?

(b) To deal with the previous problem, we will assume no three-factor interaction is present. This allows us to base our analysis on equation (7). There are a couple of ways of implementing this model in R. To be as brief as possible, if we have factors A, B, and C and response variable Y, we could code the model in R by Y ~ A + B + C + A:B + A:C + B:C. An efficient (but not obvious) way to generate the same model is Y ~ (A + B + C)^2. Substituting your variable names run this analysis and save the resulting ANOVA table. (Note: You want to save tables output in R in a monospacedfont, such as Courier New, otherwise numbers in columns won’t line up properly and your output will look like crap).

(c) Recall that R automatically treats all factors as being fixed effects factors. If we were to treat all factors as fixed effects factors, what conclusions would you draw from the ANOVA table? We are looking for a brief summary.

(d) There is a good argument that Dentist would be better considered as a random effects factor. What would we be assuming if we did so?

(e) If we consider Dentist as a random effects factor, what adjustments need to be made to the ANOVA table? Be precise; we are looking for changes in F-ratios (specify how these are calculated) and associated p-values.

(f) Do the conclusions change as a result of treating Dentists as a random effects factor? If so, how?

(g) You should have found one significant two-factor interaction. Prepare two side-by-side or top/bottom interaction plots that illustrate the nature of this interaction (these will look best if you expand the side-by-side plots to “full screen” before saving the pdf). What appears to be the cause of the interaction?

(h) On the basis of the analysis conducted what recommendation should you make regarding preparing the dental gold alloy for fillings, and why?

(i) The effect (or lack of effect) of temperature on hardness can be examined graphically use the plot(Y ~ X) command, substituting Hardness as the response variable and Temperature as the predictor variable. How does the graphical output differ if Temperature is declared as a factor, in comparison to if Temperature is treated as a numeric variable (previously stored as Temp.Numeric)? How would you describe the effect of Temperature on Hardness?

dental-gold-hardness-subset-stat2050-w21-5yvb2kzv-5ooyx3hn.csv