Question

Select at least three variables that you believe have a linear relationship.

A. Specify which variable is dependent and which are independent.

B. Collect the data for these variables and describe your data collection technique and why it was appropriate as well as why the sample size was best.

C. Submit the data collected by submitting the SPSS data file with your submission.

D. Find the Correlation coefficient for each of the possible pairings of dependent and independent variables and describe the relationship in terms of strength and direction.

E. Find a linear model of the relationship between the three (or more) variables of interest.

F. Explain the validity of the model.

Answer 1

temperature is dependent variable, latitude, longitude and elevation are independent

Source Sum square DF Mean square

Total 181439 1069 170

Error 25301 1066 24

Regression 156138 3 52046

F = 52046/24 ≈ 2169 on 3,1066 DF

The relative importance of the variables can be assessed based on the PVE’s for various submodels:

Predictors PVE F

Latitude 0.75 1601

Longitude 0.10 59

Elevation 0.02 9

Longitude, Elevation 0.19 82

Latitude, Elevation 0.75 1080

Latitude, Longitude 0.85 2000

Latitude, Longitude, Elevation 0.86 1645

Latitude is by far the most important predictor, with longitude a distant second.

Up to this point, each predictor variable has been incorporated into the regression function through an additive term βiXi . Such a term is called a main effect.

r the temperature data, each of the three possible interactions was added (individually) to the model along with the three main effects. PVE’s and F statistics are given below:

Interactions PVE F

Latitude×Longitude 0.88 1514

Latitude×Elevation 0.86 1347

Longitude×Elevation 0.88 1519

The coefficients for the model including the latitude×longitude interaction are:

E(Y |X) = 188 − 4.25Latitude + 0.61Longitude − 0.003Elevation + 0.02Latitude × Longitude

Longitude ranges from 68◦ to 125◦ in this data set.

Thus in the western US the model can be approximated as E(Y |X) ≈ 264 − 1.75Latitude − 0.003Elevation,

while in the eastern US, the model can be aproximated as E(Y |X) ≈ 229 − 2.89Latitude − 0.003Elevation.

This tells us that the effect of latitude was stronger in the eastern US than in the western US.