[2021] STAT2401: Analysis of Experiments - Q1 Correlation Matirx, F-test and P-value
1. The data watershd-Version-0.txt is a set of simulated data on peak rate of flow Q (in cubic feet per second (cfs)) of water from six watersheds following storm episodes. The storm episodes have been chosen from a larger data set to give a range of storm intensities. The independent variables are
Variable Definition
- X1 Area of watershed (m2)
- X2 Area impervious to water (m2)
- X3 Average slope of watershed (percent)
- X4 Longest stream flow in watershed (thousands of feet)
- X5 Surface absorbency index, 0 = complete absorbency, 100 = no absorbency
- X6 Estimated soil storage capacity (inches of water)
- X7 Infiltration rate of water into soil (inches/hour)
- X8 Rainfall (inches)
- X9 Time period during which rainfall exceeded 14 inch/hr.
This dataset watershd-Version-0.txt is available in LMS unit STAT2401 on- line. Please download the data, save it in “your working directory”, and read in the data by
setwd("your working directory")
watershd = read.table(file="watershd-Version-0.txt",header=T)
(a) Take Q as the dependent variable and X1-X9 as independent variables.
- (i) Compute the correlation matrix for all variables including the dependent variable Q (Report the R code you use). By inspection of the correla- tions determine which variable with highest correlation is most likely to contribute significantly to variation in Q (Report the variable and the correlation). [3 marks]
- (ii) Compute the correlation matrix using the logarithms of all variables including the dependent variable (Report the R code you use). How does this change the correlations and your conclusions (from part (i)) about which variable is most likely to contribute significantly to variation in log(Q) (Report the variable and the correlation)? [3 marks]
(b) Describe the process of backwards variable selection, implemented using F- test and p-value approach, for a multiple linear regression model.
[5 marks]
(c) Fit a linear model using log(Q) as the dependent variable and using the logarithms of all nine of the original independent variables as the new inde- pendent variables. Write down your fitted model equation, R code used to fit this model, and F-statistic with associated p-value from the information below the summary table of coecients of this fitted model. [4 marks]
(d) Based on the model fitted in part (c), is log(Q) related to at least some of the explanatory variables? Explain your reasoning. [3 marks]
(e) Starting with the NULL model, i.e. log(Q) = 0 + ✏, perform forward vari- able selection using F-tests to select a model. The variables we consider are the logarithms of all nine independent variables. Work at a 5% significance level. Write down your fitted model equation for the model finally selected. [5 marks]
(f) Starting with the FULL model, i.e. regress log(Q) on the logarithms of all nine independent variables (this is the model in part (c)), perform backward variable selection using F-tests to select a model. Work at a 5% significance level. Write down your fitted model equation for the model finally selected. [5 marks]
(g) State the common explanatory variables of the final models found in parts (e) and (f). [3 marks]
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