A data set obtained to study lung function includes the following variables.

A data set obtained to study lung function includes the following variables.

complete solutions correct answers key 

                                      pemax:     maximum expiration pressure (cm H2O)

                                      height:  height in centimeters    

                                      weight:  weight in kilograms

                                      bmp:          body mass (weight/height²) as percent of  median for age

    The goal was to model pemax in terms of the three other variables.  Here is the scatterplot matrix of the data,

    including histograms of each variable and pairwise least squares fits.

    The first model examined included all three predictors. Ch, Cw, and Cb are symbols for the partial slopes.

         Model 1: pemax = intercept + Ch*height + Cw*weight + Cb*bmp

            Estimate Std. Error t value Pr(>|t|) 

(Intercept) 245.3936   119.8927   2.047   0.0534

height       -0.8264     0.7808  -1.058   0.3019 

weight        2.7717     1.1377   2.436   0.0238

bmp          -1.4876     0.7375  -2.017   0.0566

Residual standard error: 25.24 on 21 degrees of freedom

R-squared:  0.5015,    Adjusted R-squared:  0.4302

F-statistic: 7.041 on 3 and 21 DF,  p-value: 0.00187

    a. Explain what can be concluded from the F test. Include a statement of the null hypothesis..

    b. What can be concluded from the t tests and why?

Three additional models were also examined.

    Model 2: pemax = intercept + Cw*weight + Cb*bmp  

                         Estimate Std. Error t value Pr(>|t|)   

 (Intercept) 124.8297    37.4786   3.331  0.00303 

  weight       1.6403     0.3900   4.206  0.00037

  bmp         -1.0054     0.5814  -1.729  0.09780  

  Residual standard error: 25.31 on 22 degrees of freedom

  R-squared:  0.4749,    Adjusted R-squared:  0.4271

  F-statistic: 9.947 on 2 and 22 DF,  p-value: 0.0008374

    Model 3: pemax = intercept + Cw*weight

                             Estimate Std. Error t value Pr(>|t|)   

  (Intercept)  63.5456    12.7016   5.003 4.63e-05

  weight        1.1867     0.3009   3.944 0.000646

  Residual standard error: 26.38 on 23 degrees of freedom

  R-squared:  0.4035,    Adjusted R-squared:  0.3776

  F-statistic: 15.56 on 1 and 23 DF,  p-value: 0.0006457

          Model 4: pemax = intercept + Cb*bmp

                           Estimate Std. Error t value Pr(>|t|)

 (Intercept)  59.0802    44.7444   1.320     0.20

 bmp           0.6392     0.5652   1.131     0.27

 Residual standard error: 33.24 on 23 degrees of freedom

 R-squared:  0.05268,   Adjusted R-squared:  0.01149

 F-statistic: 1.279 on 1 and 23 DF,  p-value: 0.2698

    c. Based on the regression summaries of Models 1 and 2, which one appears to be the better model? Explain,

        including brief discussion of the p-values, Residual standard error, and Adjusted R-squared, also

        explaining how the latter differs from R-squared.

    d. For Model 2, explain the meanings of the partial slopes for weight and for bmp, in context, in terms of

        pemax, weight, and bmp, including correct units.

    e. Are Models 3 and 4 consistent with Model 2? How, in particular, can it make sense that Model 4 has a

        positive slope for bmp, while Model 2 has a negatuve partial slope for bmp, as well as a much smaller

        p-value for that partial slope? The plot on the next page may help explain the answer.