A university medical center urology group (Stamey, *et al.*, 1989) was interested in the association between a prostate-specific antigen (PSA) and a number of prognostic clinical measurements in men with advanced prostate cancer. Data were collected on 97 men who were about to undergo radical prostectomies. We will look at a subset of this dataset for this assignment; later in the semester we’ll be able to analyze the full complex dataset.

column | Variable name | Description |
---|---|---|

1 | Identification number | 1-97 |

2* | PSA level | Serum prostate-specific antigen level (ng/ml) |

3* | Cancer volume | Estimate of prostate cancer volume (cc) |

4* | Weight | Prostate weight (gm) |

5* | Age | Age of patient (years) |

6 | Benign prostatic | Amount of benign prostatic hyperplasia (cm2) hyperplasia |

7 | Seminal vesicle invasion | Presence or absence of seminal vesicle invasion: 1 if yes; 0 if no |

8 | Capsular penetration | Degree of capsular penetration (cm) |

9 | Gleason score | Pathologically determined grade of disease (6,7,8), higher indicates worse prognosis |

**Background:** Until recently, PSA was commonly recommended as a screening mechanism for detecting prostate cancer. To be an efficient screening tool it is important that we understand how PSA levels relate to factors that may determine prognosis and outcome. The PSA test measures the blood level of prostate-specific antigen, an enzyme produced by the prostate. PSA levels under 4 ng/mL (nanograms per milliliter) are generally considered normal, while levels over 4 ng/mL are considered abnormal (although in men over 65 levels up to 6.5 ng/mL may be acceptable, depending upon each laboratorys reference ranges). PSA levels between 4 and 10 ng/mL indicate a risk of prostate cancer higher than normal, but the risk does not seem to rise within this six-point range. When the PSA level is above 10 ng/mL, the association with cancer becomes stronger. However, PSA is not a perfect test. Some men with prostate cancer do not have an elevated PSA, and most men with an elevated PSA do not have prostate cancer. PSA levels can change for many reasons other than cancer. Two common causes of high PSA levels are enlargement of the prostate (benign prostatic hypertrophy (BPH)) and infection in the prostate (prostatitis).

A goal here is to build a multiple regression model to predict PSA level `PSA`

from the cancer volume `V`

, Prostate weight `Wt`

, and Age `Age`

. A reasonable strategy would be to:

- Examine the relationship between the response and the potential predictors.
- Decide whether any of the variables should be transformed.
- Perform a backward elimination using the desired response and predictors.
- Given the selected model, examine the residuals and check for influential cases.
- Repeat the process, if necessary.
- Interpret the model and discuss any model limitations.

*Describe any patterns you see in the data. Are the ranges for each variable reasonable? Extreme/unusual observations? Strong nonlinear trends with the response suggesting a transformation?*

```
library(tidyverse)
# load ada functions
source("ada_functions.R")
# First, download the data to your computer,
# save in the same folder as this Rmd file.
# read the data
<- read_csv("ADA2_CL_06_psa.csv")
dat_psa str(dat_psa)
```

```
tibble [97 x 9] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$ ID : num [1:97] 1 2 3 4 5 6 7 8 9 10 ...
$ PSA : num [1:97] 0.651 0.852 0.852 0.852 1.448 ...
$ cancer_volume : num [1:97] 0.56 0.372 0.601 0.301 2.117 ...
$ prostate_weight : num [1:97] 16 27.7 14.7 26.6 30.9 ...
$ patient_age : num [1:97] 50 58 74 58 62 50 64 58 47 63 ...
$ benign_prostatic_hyperplasia: num [1:97] 0 0 0 0 0 ...
$ seminal_vesicle_invasion : num [1:97] 0 0 0 0 0 0 0 0 0 0 ...
$ capsular_penetration : num [1:97] 0 0 0 0 0 0 0 0 0 0 ...
$ gleason_score : num [1:97] 6 7 7 6 6 6 6 6 7 6 ...
- attr(*, "spec")=
.. cols(
.. ID = col_double(),
.. PSA = col_double(),
.. cancer_volume = col_double(),
.. prostate_weight = col_double(),
.. patient_age = col_double(),
.. benign_prostatic_hyperplasia = col_double(),
.. seminal_vesicle_invasion = col_double(),
.. capsular_penetration = col_double(),
.. gleason_score = col_double()
.. )
```

```
#dat_psa
<-
dat_psa %>%
dat_psa # select variables we want to use for this assignment
select(
ID
, PSA
, cancer_volume
, prostate_weight
, patient_age%>%
) # simplify column names
rename(
PSA = PSA
V = cancer_volume
, Wt = prostate_weight
, Age = patient_age
,
)
str(dat_psa)
```

```
tibble [97 x 5] (S3: tbl_df/tbl/data.frame)
$ ID : num [1:97] 1 2 3 4 5 6 7 8 9 10 ...
$ PSA: num [1:97] 0.651 0.852 0.852 0.852 1.448 ...
$ V : num [1:97] 0.56 0.372 0.601 0.301 2.117 ...
$ Wt : num [1:97] 16 27.7 14.7 26.6 30.9 ...
$ Age: num [1:97] 50 58 74 58 62 50 64 58 47 63 ...
```

`summary(dat_psa)`

```
ID PSA V Wt Age
Min. : 1 Min. : 0.651 Min. : 0.2592 Min. : 10.70 Min. :41.00
1st Qu.:25 1st Qu.: 5.641 1st Qu.: 1.6653 1st Qu.: 29.37 1st Qu.:60.00
Median :49 Median : 13.330 Median : 4.2631 Median : 37.34 Median :65.00
Mean :49 Mean : 23.730 Mean : 6.9987 Mean : 45.49 Mean :63.87
3rd Qu.:73 3rd Qu.: 21.328 3rd Qu.: 8.4149 3rd Qu.: 48.42 3rd Qu.:68.00
Max. :97 Max. :265.072 Max. :45.6042 Max. :450.34 Max. :79.00
```

```
library(ggplot2)
library(GGally)
#p <- ggpairs(dat_psa)
# put scatterplots on top so y axis is vertical
<- ggpairs(dat_psa %>% select(PSA, V, Wt, Age)
p #, upper = list(continuous = wrap("points", alpha = 0.2, size = 0.5))
upper = list(continuous = "points")
, lower = list(continuous = "cor")
,
)print(p)
```

```
# correlation matrix and associated p-values testing "H0: rho == 0"
#library(Hmisc)
%>%
dat_psa select(PSA, V, Wt, Age) %>%
as.matrix() %>%
::rcorr() Hmisc
```

```
PSA V Wt Age
PSA 1.00 0.62 0.03 0.02
V 0.62 1.00 0.01 0.04
Wt 0.03 0.01 1.00 0.16
Age 0.02 0.04 0.16 1.00
n= 97
P
PSA V Wt Age
PSA 0.0000 0.7988 0.8672
V 0.0000 0.9604 0.7038
Wt 0.7988 0.9604 0.1078
Age 0.8672 0.7038 0.1078
```

Bivariate scatterplots with points labeled by ID number to help characterize relationships and identify outliers.

[answer]

Fit an appropriate full model and perform backward selection using **BIC**. Discuss the diagnostics in terms of influential observations or problematic structure in the residuals.

[answer]

Below I’ll get you started with the linear model with all the selected main effects and a set of diagnostic plots.

```
# fit full model
<-
lm_psa_full lm(
~ V + Wt + Age
PSA data = dat_psa
,
)
#library(car)
#Anova(aov(lm_psa_full), type=3)
summary(lm_psa_full)
```

```
Call:
lm(formula = PSA ~ V + Wt + Age, data = dat_psa)
Residuals:
Min 1Q Median 3Q Max
-61.687 -8.720 -1.619 3.223 181.468
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.05753 28.61023 0.142 0.888
V 3.23156 0.41936 7.706 1.39e-11 ***
Wt 0.02220 0.07325 0.303 0.762
Age -0.06191 0.45001 -0.138 0.891
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 32.36 on 93 degrees of freedom
Multiple R-squared: 0.3902, Adjusted R-squared: 0.3705
F-statistic: 19.84 on 3 and 93 DF, p-value: 5.045e-10
```

Model selection starts here.

If the model doesn’t fit well (diagnostics tell you this, not \(R^2\) or significance tests), then address the lack of model fit. Transformations and removing influential points are two strategies. The decisions you make should be based on what you observed in the residual plots. If there’s an influential observation, remove it and see how that affects the backward selection (whether the same predictors are retained), the model fit (diagnostics), and regression coefficient estimates (betas). If there’s a pattern in the residuals that can be addressed by a transformation, guess at the appropriate transformation and try it.

Repeat until you are satisfied with the diagnostics meeting the model assumptions. Below, briefly outline what you did (no need to show all the output) by (1) identifying what you observed in the diagostics and (2) the strategy you took to address that issue. Finally, show the final model and the diagnostics for that. Describe how the final model is different from the original; in particular discuss whether variables retained are different from backward selection and whether the sign and magnitude of the regression coefficients are much different.

```
# In the diagnostic plots, R uses the row label as the "Obs. number"
# Thus, you need to remove observations by their ID number.
# (An ID doesn't change when a lower row is removed, but the Obs. number will.)
# Below is an example of how to do that.
# remove influential observations
dat_psa_sub <-
dat_psa %>%
filter(
!(ID %in% c( [ID numbers here] )
)
```

[answer]

You may want to use the 3D visualization for your original final model and after making some decisions based on diagnostics. Are you convinced it’s a reasonable summary of the general relationship?

```
## visualize in 3D with surface
# library(rgl)
# library(car)
# scatter3d(y ~ x1 + x2, data = dat)
```

What proportion of variation in the response does the model explain over the mean of the response? (This quantity indicates how precisely this model will predict new observations.)

[answer]

Write the equation for the final model and interpret each model coefficient.

[answer]

To which population of people does this model make inference to? Go back to the abstract of the original study (first sentence) to see which population this sample of men was drawn from.

[answer]