Chapter 13 Regression designs

## Parsed with column specification:
## cols(
##   Cholesterol = col_double(),
##   Age = col_double(),
##   Sex = col_character(),
##   SBP = col_double(),
##   DBP = col_double(),
##   CIG = col_double()
## )

DASL

The Framingham Heart Study [PDF] is one of the longest running health studies. It has followed original subjects, their children, and their grand children, looking for factors that affect cardiac health. These data only include subjects whose cholesterol was measured in the first exam.

Source: “Statistical Methods in Epidemiology” by H.A. Kahn and C.T. Sempos

For ease of use, the data linked below are a random selection of 500 from the total 1406 cases in the DASL data set.

13.1 Description

For the continuous tables, we present descriptive statistics.

## Non-numerical variable(s) ignored: Sex

	Age	Cholesterol	CIG	DBP	SBP
Mean	52.26	233.98	8.26	90.40	148.17
Std.Dev	4.78	47.93	11.39	13.72	27.84
Min	45.00	121.00	0.00	52.00	90.00
Median	52.00	230.00	0.00	90.00	142.00
Max	62.00	430.00	50.00	140.00	290.00
N.Valid	500.00	500.00	500.00	500.00	500.00
Pct.Valid	100.00	100.00	100.00	100.00	100.00

For the discrete variables, we present frequencies.

	Freq	% Valid	% Valid Cum.	% Total	% Total Cum.
FEM	258	51.60	51.60	51.60	51.60
MALE	242	48.40	100.00	48.40	100.00
<NA>	0			0.00	100.00
Total	500	100.00	100.00	100.00	100.00

13.2 Visualization

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Figure 13.1: Relationships between key variables.

13.3 Regression analysis

Model statistics


Multiple \(R^2\)	0.049
Multiple \(R^2_{adj}\)	0.040
\(F_{5,494}\)	5.128
Sig. (\(p\))	0.000
Resid. SD	46.972
Resid. df	494

Regression coefficients

	Coef.	Std. Err.	\(t\) stat.	Sig. (\(p\))	VIF
(Intercept)	175.671	28.784	6.103	0.000
Age	0.668	0.465	1.434	0.152	1.120
SexMALE	−18.924	4.575	−4.136	0.000	1.185
SBP	−0.037	0.120	−0.307	0.759	2.523
DBP	0.413	0.241	1.711	0.088	2.481
CIG	0.082	0.202	0.407	0.684	1.196

13.3.1 Model checking

13.3.2 Sequential model testing

Model	Resid. SS	Resid. df	Add. SS	Add. df	\(F\)	Sig. (\(p\))	AIC
Cholesterol ~ 1	1,146,509.712	499					5,291.751
Cholesterol ~ Age	1,144,509.033	498	2,000.679	1	0.907	0.341	5,292.878
Cholesterol ~ Age + Sex	1,102,051.249	497	42,457.783	1	19.243	0.000	5,275.976
Cholesterol ~ Age + Sex + SBP + DBP	1,090,300.348	495	11,750.901	2	2.663	0.071	5,274.616
Cholesterol ~ Age + Sex + SBP + DBP + CIG	1,089,935.082	494	365.266	1	0.166	0.684	5,276.449

Statistical models
	Intercept	+Age	+Sex	+BPr	+Cig
(Intercept)	233.98^***	212.09^***	214.46^***	178.16^***	175.67^***
	(2.14)	(23.56)	(23.14)	(28.10)	(28.78)
Age		0.42	0.54	0.63	0.67
		(0.45)	(0.44)	(0.46)	(0.47)
Sex (Male)			-18.48^***	-18.22^***	-18.92^***
			(4.22)	(4.23)	(4.58)
Sys. Bld Pres.				-0.04	-0.04
				(0.12)	(0.12)
Dia. Bld Pres.				0.41	0.41
				(0.24)	(0.24)
Cig/day					0.08
					(0.20)
R²	0.00	0.00	0.04	0.05	0.05
Adj. R²	0.00	-0.00	0.03	0.04	0.04
Num. obs.	500	500	500	500	500
^*p < 0.001; ^p < 0.01; ^*p < 0.05