attached the files
Problem 4.1 Problem 4.1 Consider the experiments on plant growth in environmentally-controlled chambers that were described in Problem 2.3. Suppose we wish to test the effects of two factors on plant growth: CO2 concentration (ambient vs. high) and temperature (low vs. high). These factors can be varied across the chambers, but a single chamber must be kept at a given CO2 concentration and temperature for the duration of the experiment. Assume we have 12 chambers at our disposal; each chamber can accommodate 12 seedlings; and we have identified 144 healthy seedlings of nearly uniform weight. The response in each chamber will be summarized as the mean accumulated dry weight of the 12 plants in that chamber after three months. 1. Describe how you would design and execute an experiment to test the effects of CO2 concentration and temperature on plant growth. 1. Identify the statistical procedure you would use to analyze the data, and provide an ANOVA table that shows the sources of variation and associated degrees of freedom. Problem 4.2 Problem 4.2 Consider the data set chem, on the yield of a chemical reaction as a function of the type of base and the type of alcohol used in the reaction. See script2 for some R tips, and show any R code you use to answer the following questions. CHEM - A chemical production process consists of a first reaction with an ALCOHOL and a second reaction with a BASE. A 3x2 factorial experiment was conducted with four replicate reactions for each of the six treatment combinations. The response is percent yield. Note that ALCOHOL and BASE are both categorical variables, i.e., they are not ordered in any way. 1. Draw a boxplot of yield for the six treatment combinations. 1. Fit the two-way ANOVA model, including the interaction term. Show the output, and interpret the results. 1. Create a table of the cell means (i.e., the mean yields for each of the six treatment combinations). 1. Calculate the simple effects of base (2 minus 1) for the three levels of the alcohol factor. Does there appear to be an interaction between base and alcohol? Explain. 1. Let be the mean yield for the combination of base and alcohol (; ). Estimate, and compute a confidence interval for, the following contrast: What if anything does this interval tell you about the interaction between base and alcohol? 1. Using the residuals from the two-way ANOVA: (i) draw and interpret a plot to assess the normality assumption; (ii) draw and interpret a plot to assess the constant-variance assumption; and (iii) perform and interpret the results of Levene’s test. Problem 4.3 Problem 4.3 The data set plant is from a factorial experiment on the effects of CO2 concentration (ambient vs. high) and soil temperature (low vs. high) on plant growth in 12 environmentally-controlled chambers. The four treatment combinations were randomly assigned to the chambers (3 chambers per treatment combination), and 12 plants were grown in each chamber. The variable plant.wt gives the mean accumulated dry weights for the 12 plants in each chamber. This experiment was actually run on three plant species simultaneously. For this problem, use only the data for needlegrass (species 2). One way to extract data for species 2: ## Load the full data set plant = read.table("../data/plant.txt", header=TRUE); ## Extract data for species 2 ind = plant$species==2; plant.new = plant[ind,]; ## Inspect the data structure str(plant.new); plant.new; ## Attach the data frame attach(plant.new); ## Additional preparation, plot the data, etc. ## ... Note: When converting temp to a factor variable, one can change the level labels to make the data more readable. (This is not required for this homework.) temp.f = factor(temp, levels = c(0, 1), labels=c("13 C", "18 C")); data.frame(temp, temp.f) co2.f = factor(co2, levels = c(0, 1), labels = c("ambient", "high")); data.frame(co2, co2.f) 1. Draw a graph of the plant weights in the four combinations of CO2 and temperature, and describe the patterns you see. 1. Perform an analysis of variance of plant weight, using a model that includes co2, temp, and their interaction. Include (i) a summary of the mean response in the different treatment combinations; (ii) the ANOVA table; and (iii) an interpretation of your results. Note: For this part, use ANOVA, rather than regression, output. There is not an exact correspondence between the P-values in the regression output and those in the ANOVA output, as explained in the lecture notes. 1. Use Levene’s test to evaluate the constant-variance assumption of your ANOVA model. Is the result consistent with what you see in your graph from part (a)? Explain. 1. Since the interaction term is nonsignificant, drop it and fit a model with just the main effects of CO2 and temperature. Present both the ANOVA and regression output, and explain what they suggest about the effects of CO2 and temperature on plant weight. 1. Using the ANOVA output from the two models you fit above, do an extra-sum- of-squares test of the null hypothesis that there is no interactive effect of CO2 and temperature on plant growth. That is, extract the needed sums of squares and degrees of freedom, and plug them in to the general formula for the extra-sum-of-squares F- test. Show how to obtain the P-value, and verify that it is the same as that for the interaction term in the full model from part (b). Problem 4.4 Problem 4.4 Consider the anode data set, on the effect of furnace position and temperature on carbon anode density. ANODE - a two-factor factorial experiment on the effects of TEMPERATURE and POSITION in a furnace on carbon anode DENSITY. There were three furnaces, each kept at a particular temperature (the fixed effect of interest). In each furnace, there were two positions (consider this to be a random effect, crossed with temperature). Eighteen anodes (the experimental units) were randomly assigned to the six temperature-position combinations, three anodes (replicates) per combination. The response of interest is the density of an anode after a period of incubation in the furnace. Suppose that the positions used are a random sample from a set of possible positions (i.e., position is a random effect), but the investigators are interested in the particular temperatures used in the experiment (i.e., temperature is a fixed effect). Assume the constant-variance assumption is satisfied for these data. (In fact, Levene’s test on the six treatment combinations gives .) 1. Prepare a graphical summary of anode density in the six treatment combinations. Comment on the apparent main effects of position and temperature, and the interaction between them. 1. Perform an appropriate analysis of variance on these data, modeling both position and temperature as categorical variables. Show the ANOVA table; perform F-tests for both main effects and their interaction; and interpret your results. Do not drop the interaction term, even though it is nonsignificant. Note: Fit a fixed-effects model and then re-do the -tests by hand if necessary, as explained on pp. 239-240 of Kuehl. 1. A colleague suggests that, since temperature is measured on a continuous scale, it would make more sense to model temperature as a quantitative variable, in an ANCOVA-like approach, instead of as a qualitative (categorical) variable. Eager to please, you proceed as follows. 1. Fit a model of density as a function of the quantitative temperature variable and a factor for position, without the interaction term (since it is nonsignificant). Hand in the regression output. 1. Interpret the magnitude and the statistical significance of the regression coefficent for temperature. How does your conclusion compare to what you inferred from the earlier model that used a categorical version of temperature? Do you agree with your colleage that temperature should be treated as a quantitative predictor? Explain.