BIOST/STAT 578B – Data Analysis



BIOST/STAT 579 – Data Analysis

Longitudinal Data Analysis

Longitudinal data present special opportunities and challenges that don’t exist with cross-sectional data analysis. First, there are questions that can be addressed with longitudinal data that cannot even be considered with cross-sectional data. For example, with longitudinal data one can distinguish between cross-sectional and longitudinal associations. A second feature of longitudinal data is that there are three distinct types of models that can be considered: 1) marginal models, 2) random-effects models, and 3) transition models. These models address different scientific questions so the choice of model type is extremely important and needs to be made based on the scientific question of interest rather than on statistical considerations. The third issue to be considered is modeling the correlation structure. With longitudinal data there is a wide variety of correlation structures to choose from.

Outline

1. Longitudinal versus cross-sectional associations

2. Contrasts between three approaches to modeling longitudinal data

3. Modeling longitudinal correlation structures

1. Longitudinal versus cross-sectional associations

Example I (Potthoff & Roy growth data):

Distances between two facial landmarks for 27 children (11 females, 16 males) at ages 8, 10, 12, and 14 years. Spaghetti plots of the data are shown below. (Note: age is centered at 11.)

[pic]

Recall that a linear model with gender, age, and gender-age interaction gave the same coefficients as separate analyses of subject means and individual regression slopes on age.

Linear models: (Estimate ( SE)

|Variable |Linear model on raw data (Estimate (|Linear model on means (Estimate ( |Linear model on slopes (Estimate ( |

| |Robust SE) |SE) |SE) |

|Intercept |22.6 ( 0.61 |22.6 ( 0.59 | |

|Male |2.32 ( 0.75 |2.32 ( 0.76 | |

|Age |0.48 ( 0.06 | |0.48 ( 0.10 |

|Male*Age |0.30 ( 0.12 | |0.30 ( 0.13 |

For these data the overall linear model gives the same results as the analyses of subject-specific means and slopes. This means that the age coefficient in the overall model represents the change in mean corresponding to aging of an individual child. This property results from the fact that the data are balanced. Let’s try a different data set to see what happens if the data are not balanced.

Example II (Tooth loss in periodontitis patients):

A cohort study was done on the effectiveness of treatment of periodontitis. The study enrolled approximately 800 patients with periodontitis and followed them for up to 10 years. The outcome variable is the number of teeth extracted in each follow-up year.

Results of the three analyses as above:

|Variable |Linear model on raw data (Estimate ( |Linear model on means (Estimate ( SE) |Linear model on slopes (Estimate ( SE) |

| |robust SE) | | |

|Intercept |0.181 ( 0.019 |0.187 ( 0.022 | |

|Male |0.022 ( 0.030 |0.012 ( 0.030 | |

|Age |-0.001 ( 0.002 |0.000 ( 0.003* |-0.060 ( 0.016 |

|Male X Age |0.001 ( 0.004 |0.002 ( 0.004* |0.029 ( 0.022 |

*Note that we can fit age and male X age terms to the subject means because the average value of age varies across subjects (unlike the Potthoff-Roy data).

The results show there is a longitudinal association (within subjects) between extractions and age but no cross-sectional association (between subjects). (The overall analysis gives results similar to the between subjects analysis in this case.) The analysis of slopes is not very precise because each slope is given equal weight regardless of its variability. A better analysis, which allows simultaneous estimation of cross-sectional and longitudinal associations is between and within-cluster regression, based on fitting the following model:

E(Y) = Intercept + Male * { Ave. Age + (Age – Ave. Age) }

|Variable |Linear model (Estimate ( Robust SE) |

|Intercept |0.180 ( 0.019 |

|Male |0.022 ( 0.030 |

|Ave. Age |0.002 ( 0.003 |

|Age Diff. |-0.025 ( 0.009 |

|Male X Ave. Age |0.000 ( 0.004 |

|Male X Age Diff. |0.014 ( 0.012 |

Note that the overall estimates for Age and Male X Age (from previous table) are intermediate between the cross-sectional and longitudinal estimates, but are closer to the cross-sectional estimates.

Alternative parametrization:

E(Y) = Intercept + Male*{ Baseline Age + Year }

(Note: Year = Age – Baseline Age)

| |Estimate ( Robust SE |

|Variable | |

|Intercept |0.072 ( 0.151 |

|Male |0.056 ( 0.214 |

|Baseline Age |0.002 ( 0.003 |

|Year |-0.022 ( 0.008 |

|Male X BL Age |-0.001 ( 0.004 |

|Male X Year |0.015 ( 0.011 |

Between and within-regression for a log-linear model:

log(E(Y)) = Intercept + Male * { Baseline Age + Year }

Estimates and robust SEs:

|Variable |Estimate ( Robust SE |

|Intercept |-2.362 ( 0.847 |

|Male |0.396 ( 1.128 |

|Baseline Age |0.012 ( 0.016 |

|Year |-0.130 ( 0.049 |

|Male X Baseline Age |-0.005 ( 0.022 |

|Male X Year |0.092 ( 0.063 |

2. Contrasts between three approaches to modeling longitudinal data

Three modeling approaches:

• Marginal Model (Direct Approach)

• Random Effects Model

• Transition Model

Example I (Potthoff & Roy growth data):

Marginal Model (“direct” approach):

E(Y) = Intercept + Male + Age + Male X Age

Random Effects Model:

E(Y|U) = Intercept + Male + Age + Male X Age + U

U = random effect at subject level (iid) ~ N(0, V)

Transition Model:

E(Y|YPrevious) = Intercept + Male + Age + Male X Age + YPrevious

YPrevious = value of Y at previous time point

Results of three model fits for the Potthoff-Roy data:

| |Estimate ( SE |

| |Marginal Model* |Random- Effects Model |Transition Model* |

|Variable | | | |

|Intercept |22.6 ( 0.61 |22.6 ( 0.59 |9.57 ( 3.67 |

|Male |2.32 ( 0.75 |2.32 ( 0.76 |1.03 ( 0.51 |

|Age |0.48 ( 0.06 |0.48 ( 0.09 |0.18 ( 0.12 |

|Male X Age |0.30 ( 0.12 |0.30 ( 0.12 |0.31 ( 0.13 |

|YPrevious |NA |NA |0.60 ( 0.17 |

* SEs are robust SEs.

Note: The marginal and RE models give the same results here because the data are balanced. The transition model gives very different results from the other two models because it includes adjustment for a prior outcome.

Example II (Periodontitis study):

|Variable |Marginal Model |Random- Effects Model |Transition Model |

|Intercept |0.181 ( 0.019 |0.180 ( 0.021 |0.133 ( 0.018 |

|Male |0.022 ( 0.030 |-0.002 ( 0.003 |0.040 ( 0.028 |

|Age |-0.001 ( 0.002 |0.022 ( 0.030 | 0.001 ( 0.002 |

|Male X Age |0.001 ( 0.004 |0.002 ( 0.004 |0.001 ( 0.003 |

|YPrevious |NA |NA |( |

| |Estimate ( SE |

| |Marginal Model |Random- Effects Model |Transition Model |

|Variable | | | |

|Intercept | 0.072 ( 0.151 |0.079 ( 0.158 |0.078 ( 0.136 |

|Male |0.056 ( 0.214 |0.043 ( 0.216 |0.022 ( 0.186 |

|BL-Age |0.002 ( 0.003 |0.002 ( 0.003 |0.001 ( 0.003 |

|Year |-0.022 ( 0.008 |-0.023 ( 0.007 |0.004 ( 0.006 |

|Male X BL-Age |-0.001 ( 0.004 |0.000 ( 0.004 |0.011 ( 0.007 |

|Male X Year |0.015 ( 0.011 |0.015 ( 0.010 |0.000 ( 0.004 |

|YPrevious |NA |NA |0.006 ( 0.010 |

Note: The marginal and RE models give similar results, which is not surprising (for linear models). The transition model does not differ too much from the other two models, in this case, because the prior outcome is not strongly associated with the outcome.

Example III (Logistic regression for a cross-over trial):

Ref: Diggle et al., p.180

Marginal Model:

logit(E(Y)) = Intercept + Treatment + Period

Random Effects Model:

logit(E(Y|U)) = Intercept + Treatment + Period + U

U = random effect at subject level (iid) ~ N(0, V)

Results:

| |Marginal Model* |Random- Effects Model |

|Variable | | |

|Intercept |0.67 ( 0.29 |2.2 ( 1.0 |

|Treatment |0.57 ( 0.23 |1.8 ( 0.93 |

|Period |-0.30 ( 0.23 |-1.0 ( 0.84 |

* SEs are robust SEs.

Notes:

The estimated random effects SD is very large (5.0), which contributes to the large differences between model estimates.

Differences between coefficient estimates are due to different interpretations of the two models (as opposed to different properties of the parameter estimators).

Ratios of coefficient estimates to their SEs are similar in the two models, which is often observed.

3. Modeling longitudinal correlation structures

Example (Potthoff-Roy data):

| |Independent |Exchangeable |Autoregressive |

| | |Naive SE |Robust SE | |

|Variable |Est. | | |Est. |

|Age 8 |1 |.63 |.71 |.60 |

|Age 10 |.63 |1 |.63 |.76 |

|Age 12 |.71 |.63 |1 |.79 |

|Age 14 |.60 |.76 |.79 |1 |

The correlation structure is similar, but with very different values of the correlations, for males and females:

Males:

| |Age 8 |Age 10 |Age 12 |Age 14 |

|Age 8 |1 |.44 |.56 |.32 |

|Age 10 |.44 |1 |.39 |.63 |

|Age 12 |.56 |.39 |1 |.59 |

|Age 14 |.32 |.63 |.59 |1 |

Females:

| |Age 8 |Age 10 |Age 12 |Age 14 |

|Age 8 |1 |.83 |.86 |.84 |

|Age 10 |.83 |1 |.90 |.88 |

|Age 12 |.86 |.90 |1 |.95 |

|Age 14 |.84 |.88 |.95 |1 |

Note: these differences can be seen in the spaghetti plots.

Note: such variations in correlation structure are often ignored but can be capitalized on to increase estimation precision (Stoner and Leroux, 2002)

References

Diggle, Heagerty, Liang and Zeger (2002). Longitudinal Data Analysis.

Potthoff and Roy (1964) Biometrika (facial measurement data)

Stoner and Leroux (2002) Biometrika (improving precision compared to GEE)

Survival Analysis

Example (NHANES cardiovascular hospitalization or death):

What the data look like:

chdhosp: indicator of whether or not subject had the event of interest (cardiovascular hospitalization or death)

time: time of occurrence of the event or time of censoring (end of follow-up) whichever came first

Consider subset of subjects > 70 years of age at baseline.

n mean min max

chdhosp 1110 0.5153 0 1.0000

time 1110 9.1047 0 22.0616

table(round(chdsurv$time[chdsurv$age>70],0),chdsurv$chdhosp[chdsurv$age>70])

0 1

0 24 23

1 21 60

2 20 40

3 27 41

4 30 33

5 20 47

6 28 33

7 15 22

8 27 33

9 40 28

10 29 25

11 19 42

12 22 21

13 23 31

14 27 21

15 19 22

16 23 10

17 12 16

18 12 12

19 32 10

20 38 2

21 28 0

22 2 0

Note: the above give a picture of the data but are not descriptives you would report.

Descriptive statistics to report: Kaplan-Meier estimate of the survival curve, which is the probability of “surviving” (being event-free) to a given time.

[pic]

How Kaplan-Meier is calculated (first 10 values, females, n=597):

Time Risk Event Survival SE

0.0000 597 0 1.0000 0.0000

0.0274 590 0 1.0000 0.0000

0.0548 589 1 0.9983 0.0017

0.1150 588 1 0.9966 0.0024

0.1506 587 1 0.9949 0.0029

0.1725 586 0 0.9949 0.0029

0.1862 585 1 0.9932 0.0034

0.1916 584 0 0.9932 0.0034

0.2108 583 1 0.9915 0.0038

0.4408 582 0 0.9915 0.0038

Regression analysis

The Cox proportional hazards model is the most popular model (like logistic regression for binary data there are alternatives which don’t seem to be used much). The Cox model is a linear model for the log of the hazard rate (instantaneous probability of failure given survival to a given time).

Example: Comparison of hazards for males and females among subjects greater than 70 years of age at baseline.

coxph(formula = Surv(time, chdhosp) ~ gender, data = chdsurv, subset = (age > 70))

n= 1110

coef exp(coef) se(coef) z p

gendermale 0.309 1.36 0.0843 3.67 0.00024

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.36 0.734 1.16 1.61

The hazard for cardiovascular hospitalization or death is 1.36 times as high for males as for females. This hazard ratio is highly statistically significant (significantly different from 1).

Example: Apply Cox regression to the HW problem on comparing male and female risk of CHD.

summary(coxph( Surv(time,chdhosp) ~ gender, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender, data = chdsurv)

n= 11313

coef exp(coef) se(coef) z p

gendermale 0.708 2.03 0.0388 18.3 0

exp(coef) exp(-coef) lower .95 upper .95

gendermale 2.03 0.493 1.88 2.19

Rsquare= 0.029 (max possible= 0.986 )

Likelihood ratio test= 331 on 1 df, p=0

Wald test = 334 on 1 df, p=0

Score (logrank) test = 348 on 1 df, p=0

summary(coxph( Surv(time,chdhosp) ~ gender+age, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender + age, data = chdsurv)

n= 11313

coef exp(coef) se(coef) z p

gendermale 0.5291 1.70 0.03887 13.6 0

age 0.0765 1.08 0.00169 45.2 0

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.70 0.589 1.57 1.83

age 1.08 0.926 1.08 1.08

Rsquare= 0.238 (max possible= 0.986 )

Likelihood ratio test= 3074 on 2 df, p=0

Wald test = 2289 on 2 df, p=0

Score (logrank) test = 2965 on 2 df, p=0

summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +

physact, data = chdsurv)

n=10873 (440 observations deleted due to missingness)

coef exp(coef) se(coef) z p

gendermale 0.582057 1.790 0.040131 14.50 0.0e+00

age 0.073910 1.077 0.001734 42.62 0.0e+00

povindex -0.000685 0.999 0.000119 -5.74 9.5e-09

physactYes -0.214802 0.807 0.041118 -5.22 1.8e-07

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.790 0.559 1.654 1.936

age 1.077 0.929 1.073 1.080

povindex 0.999 1.001 0.999 1.000

physactYes 0.807 1.240 0.744 0.874

Rsquare= 0.244 (max possible= 0.986 )

Likelihood ratio test= 3035 on 4 df, p=0

Wald test = 2313 on 4 df, p=0

Score (logrank) test = 2981 on 4 df, p=0

summary(coxph( Surv(time,chdhosp) ~ gender+age+povindex+physact+priorhd*diabetes, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +

physact + priorhd * diabetes, data = chdsurv)

n=10873 (440 observations deleted due to missingness)

coef exp(coef) se(coef) z p

gendermale 0.575933 1.779 0.040236 14.31 0.0e+00

age 0.069264 1.072 0.001769 39.16 0.0e+00

povindex -0.000549 0.999 0.000118 -4.66 3.2e-06

physactYes -0.149642 0.861 0.041467 -3.61 3.1e-04

priorhdYes 1.074203 2.928 0.055510 19.35 0.0e+00

diabetesYes 0.792959 2.210 0.076011 10.43 0.0e+00

priorhdYes:diabetesYes -0.202058 0.817 0.144600 -1.40 1.6e-01

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.779 0.562 1.644 1.925

age 1.072 0.933 1.068 1.075

povindex 0.999 1.001 0.999 1.000

physactYes 0.861 1.161 0.794 0.934

priorhdYes 2.928 0.342 2.626 3.264

diabetesYes 2.210 0.453 1.904 2.565

priorhdYes:diabetesYes 0.817 1.224 0.615 1.085

Rsquare= 0.275 (max possible= 0.986 )

Likelihood ratio test= 3499 on 7 df, p=0

Wald test = 2968 on 7 df, p=0

Score (logrank) test = 4175 on 7 df, p=0

summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact + priorhd*diabetes + bpdia + I(bpdia^2) + weight + I(weight^2) + height, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +

physact + priorhd * diabetes + bpdia + I(bpdia^2) + weight

+ I(weight^2) + height, data = chdsurv)

n=10815 (498 observations deleted due to missingness)

coef exp(coef) se(coef) z p

gendermale 6.11e-01 1.842 5.66e-02 10.790 0.0000

age 6.93e-02 1.072 1.87e-03 37.031 0.0000

povindex -4.59e-04 1.000 1.18e-04 -3.888 0.0001

physactYes -1.25e-01 0.883 4.17e-02 -2.998 0.0027

priorhdYes 1.04e+00 2.823 5.58e-02 18.586 0.0000

diabetesYes 7.68e-01 2.156 7.62e-02 10.089 0.0000

bpdia 5.80e-03 1.006 1.17e-02 0.495 0.6200

I(bpdia^2) 1.57e-05 1.000 6.31e-05 0.249 0.8000

weight -1.28e-02 0.987 7.49e-03 -1.705 0.0880

I(weight^2) 1.27e-04 1.000 4.45e-05 2.846 0.0044

height -6.86e-01 0.504 3.39e-01 -2.025 0.0430

priorhdYes:diabetesYes -1.90e-01 0.827 1.45e-01 -1.313 0.1900

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.842 0.543 1.649 2.059

age 1.072 0.933 1.068 1.076

povindex 1.000 1.000 0.999 1.000

physactYes 0.883 1.133 0.813 0.958

priorhdYes 2.823 0.354 2.530 3.149

diabetesYes 2.156 0.464 1.857 2.503

bpdia 1.006 0.994 0.983 1.029

I(bpdia^2) 1.000 1.000 1.000 1.000

weight 0.987 1.013 0.973 1.002

I(weight^2) 1.000 1.000 1.000 1.000

height 0.504 1.985 0.259 0.978

priorhdYes:diabetesYes 0.827 1.210 0.622 1.098

Rsquare= 0.281 (max possible= 0.986 )

Likelihood ratio test= 3566 on 12 df, p=0

Wald test = 2951 on 12 df, p=0

Score (logrank) test = 4191 on 12 df, p=0

summary(coxph( Surv(time,chdhosp) ~ gender + age + povindex + physact + priorhd*diabetes + hitched + breakdwn, data=chdsurv))

Call:

coxph(formula = Surv(time, chdhosp) ~ gender + age + povindex +

physact + priorhd * diabetes + hitched + breakdwn, data =

chdsurv)

n=10873 (440 observations deleted due to missingness)

coef exp(coef) se(coef) z p

gendermale 0.581903 1.789 0.040357 14.42 0.0e+00

age 0.069304 1.072 0.001771 39.14 0.0e+00

povindex -0.000532 0.999 0.000118 -4.51 6.5e-06

physactYes -0.149064 0.862 0.041507 -3.59 3.3e-04

priorhdYes 1.069120 2.913 0.055556 19.24 0.0e+00

diabetesYes 0.788180 2.199 0.076041 10.37 0.0e+00

hitchedYes 0.080791 1.084 0.089802 0.90 3.7e-01

breakdwnYes 0.208847 1.232 0.096550 2.16 3.1e-02

priorhdYes:diabetesYes -0.222180 0.801 0.144980 -1.53 1.3e-01

exp(coef) exp(-coef) lower .95 upper .95

gendermale 1.789 0.559 1.653 1.937

age 1.072 0.933 1.068 1.075

povindex 0.999 1.001 0.999 1.000

physactYes 0.862 1.161 0.794 0.935

priorhdYes 2.913 0.343 2.612 3.248

diabetesYes 2.199 0.455 1.895 2.553

hitchedYes 1.084 0.922 0.909 1.293

breakdwnYes 1.232 0.812 1.020 1.489

priorhdYes:diabetesYes 0.801 1.249 0.603 1.064

Rsquare= 0.276 (max possible= 0.986 )

Likelihood ratio test= 3505 on 9 df, p=0

Wald test = 2966 on 9 df, p=0

Score (logrank) test = 4176 on 9 df, p=0

For comparison, results from logistic and linear regression:

Logistic regression models:

round(summary(glm(chd~gender,family=binomial,data=chd))$coef,3)

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

(Intercept) -3.545 0.075 -47.276 0

gendermale 0.826 0.098 8.416 0

round(summary(glm(chd~gender+age,family=binomial,data=chd))$coef,3)

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

(Intercept) -8.745 0.353 -24.778 0

gendermale 0.575 0.101 5.724 0

age 0.090 0.005 16.943 0

round(summary(glm(chd~gender+age+povindex+physact,family=binomial,data=chd))$coef,3)

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

(Intercept) -8.066 0.378 -21.364 0.000

gendermale 0.653 0.103 6.310 0.000

age 0.085 0.005 15.546 0.000

povindex -0.001 0.000 -2.238 0.025

physactYes -0.507 0.113 -4.494 0.000

round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes,family=binomial,data=chd))$coef,3)

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

(Intercept) -7.889 0.383 -20.576 0.000

gendermale 0.601 0.106 5.652 0.000

age 0.074 0.006 13.197 0.000

povindex -0.001 0.000 -1.665 0.096

physactYes -0.347 0.116 -2.977 0.003

priorhdYes 1.520 0.122 12.491 0.000

diabetesYes 0.724 0.197 3.680 0.000

priorhdYes:diabetesYes -0.163 0.306 -0.531 0.595

round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes+bpdia+I(bpdia^2)+weight+I(weight^2)+height,family=binomial,data=chd))$coef,3)

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

(Intercept) -4.662 1.904 -2.449 0.014

gendermale 0.726 0.149 4.883 0.000

age 0.074 0.006 12.776 0.000

povindex 0.000 0.000 -1.429 0.153

physactYes -0.331 0.117 -2.828 0.005

priorhdYes 1.510 0.122 12.327 0.000

diabetesYes 0.729 0.198 3.689 0.000

bpdia -0.024 0.027 -0.863 0.388

I(bpdia^2) 0.000 0.000 0.872 0.383

weight -0.026 0.020 -1.336 0.181

I(weight^2) 0.000 0.000 1.355 0.175

height -0.763 0.880 -0.868 0.386

priorhdYes:diabetesYes -0.167 0.307 -0.544 0.586

round(summary(glm(chd~gender+age+povindex+physact+priorhd*diabetes+hitched+breakdwn,family=binomial,data=chd))$coef,3)

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

(Intercept) -8.025 0.444 -18.090 0.000

gendermale 0.609 0.107 5.701 0.000

age 0.074 0.006 13.190 0.000

povindex -0.001 0.000 -1.626 0.104

physactYes -0.348 0.117 -2.984 0.003

priorhdYes 1.514 0.122 12.416 0.000

diabetesYes 0.719 0.197 3.653 0.000

hitchedYes 0.127 0.236 0.537 0.592

breakdwnYes 0.180 0.247 0.730 0.466

priorhdYes:diabetesYes -0.167 0.307 -0.543 0.587

Linear models:

round(summary(glm(chd~gender, family=quasi(link=identity, variance="mu(1-mu)"), data=chd))$coef, 4)

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

(Intercept) 0.0281 0.0020 13.7204 0

gendermale 0.0338 0.0042 8.0374 0

round(summary(glm(chd~gender+I(age-50), start=c(.03,.02,.001), family=quasi(link=identity,variance="mu(1-mu)"), data=chd))$coef, 3)

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

(Intercept) 0.035 0.002 20.185 0

gendermale 0.014 0.003 4.815 0

I(age - 50) 0.001 0.000 20.185 0

There were 27 warnings (use warnings() to see them)

round(summary(glm(chd~gender+I(age-50) + povindex + physact, start=c(.03,.02,0,0,0),family=quasi(link=identity, variance="mu(1-mu)"), data=chd))$coef, 3)

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

(Intercept) 0.037 0.003 14.526 0.000

gendermale 0.021 0.003 6.181 0.000

I(age - 50) 0.001 0.000 8.562 0.000

povindex 0.000 0.000 -3.284 0.001

physactYes -0.006 0.003 -2.163 0.031

There were 29 warnings (use warnings() to see them)

round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes, start=c(.03,.02,0,0,0,0,0,0), family = quasi(link=identity,variance="mu(1-mu)"),data=chd))$coef,3)

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

(Intercept) 0.031 0.002 13.355 0.000

gendermale 0.019 0.003 6.269 0.000

I(age - 50) 0.001 0.000 7.864 0.000

povindex 0.000 0.000 -3.141 0.002

physactYes -0.005 0.002 -1.816 0.069

priorhdYes 0.070 0.010 6.938 0.000

diabetesYes 0.018 0.009 1.915 0.056

priorhdYes:diabetesYes 0.033 0.032 1.020 0.308

There were 29 warnings (use warnings() to see them)

round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes + bpdia + I(bpdia^2) + weight + I(weight^2) + height, start = c(.03,.02,0,0,0,0,0,0,0,0,0,0,0), family = quasi(link=identity,variance="mu(1-mu)"),data=chd))$coef,3)

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

(Intercept) 0.096 0.046 2.071 0.038

gendermale 0.022 0.004 5.676 0.000

I(age - 50) 0.001 0.000 7.427 0.000

povindex 0.000 0.000 -1.847 0.065

physactYes -0.004 0.003 -1.691 0.091

priorhdYes 0.066 0.010 6.574 0.000

diabetesYes 0.017 0.009 1.845 0.065

bpdia -0.001 0.001 -0.687 0.492

I(bpdia^2) 0.000 0.000 0.601 0.548

weight 0.000 0.000 -0.894 0.372

I(weight^2) 0.000 0.000 0.770 0.441

height -0.013 0.021 -0.648 0.517

priorhdYes:diabetesYes 0.030 0.031 0.966 0.334

There were 27 warnings (use warnings() to see them)

round(summary(glm(chd~gender+I(age-50) + povindex + physact + priorhd*diabetes + hitched + breakdwn, start = c(.03,.02,0,0,0,0,0,0,0,0), family = quasi(link=identity,variance="mu(1-mu)"),data=chd))$coef,3)

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

(Intercept) 0.030 0.005 6.444 0.000

gendermale 0.019 0.003 6.279 0.000

I(age - 50) 0.001 0.000 7.830 0.000

povindex 0.000 0.000 -3.079 0.002

physactYes -0.005 0.002 -1.816 0.069

priorhdYes 0.070 0.010 6.917 0.000

diabetesYes 0.018 0.009 1.903 0.057

hitchedYes 0.000 0.005 0.078 0.938

breakdwnYes 0.002 0.008 0.287 0.774

priorhdYes:diabetesYes 0.032 0.032 1.019 0.308

There were 29 warnings (use warnings() to see them)

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