# Applied Logistic Regression, Second Edition by Hosmer and Lemeshow Chapter 2: Multiple logistic regression

## Figure 1: Hosmer and Lemeshow data on low birthweight

page 32 Table 2.1 An example of the coding of the design variables for race, coded at three levels.

```data lowbwt1;
set 'd:hosmerdatalowbwt';
if race = 1 then do; race2 = 0; race3 = 0; end;
if race = 2 then do; race2 = 1; race3 = 0; end;
if race = 3 then do; race2 = 0; race3 = 1; end;
run;
proc print data=lowbwt1 (obs=3);
var race race2 race3;
run;```
```Obs    RACE    race2    race3

1      2       1        0
2      3       0        1
3      1       0        0```

page 36 Table 2.2 Estimated coefficients for a multiple logistic regression model using the variables
age, weight at last menstrual period (lwt), race and number of first trimester physician
visits from the low birth weight study.

NOTE: We have bolded the relevant output.

```proc logistic data=lowbwt1 descending;
model low = age lwt race2 race3 ftv;
run;
quit;The LOGISTIC Procedure                    Model Information
Data Set                      WORK.LOWBWT1
Response Variable             LOW                  {amp}lt; 2500g
Number of Response Levels     2
Number of Observations        189
Optimization Technique        Fisher's scoring          Response Profile Ordered                      Total
Value          LOW     Frequency
1            1            59
2            0           130                   Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.        Model Fit Statistics                              Intercept
Intercept         and
Criterion        Only        Covariates
AIC              236.672        234.573
SC               239.914        254.023
-2 Log L         234.672        222.573        Testing Global Null Hypothesis: BETA=0Test                 Chi-Square       DF     Pr {amp}gt; ChiSq
Likelihood Ratio        12.0991        5         0.0335
Score                   11.3876        5         0.0442
Wald                    10.6964        5         0.0577The LOGISTIC Procedure             Analysis of Maximum Likelihood Estimates                               Standard
Parameter    DF    Estimate       Error    Chi-Square    Pr {amp}gt; ChiSq
Intercept     1      1.2953      1.0714        1.4616        0.2267
AGE           1     -0.0238      0.0337        0.4988        0.4800
LWT           1     -0.0142      0.00654       4.7428        0.0294
race2         1      1.0039      0.4979        4.0660        0.0438
race3         1      0.4331      0.3622        1.4296        0.2318
FTV           1     -0.0493      0.1672        0.0869        0.7681           Odds Ratio Estimates             Point          95% Wald
Effect    Estimate      Confidence Limits
AGE          0.976       0.914       1.043
LWT          0.986       0.973       0.999
race2        2.729       1.029       7.240
race3        1.542       0.758       3.136
FTV          0.952       0.686       1.321Association of Predicted Probabilities and Observed Responses
Percent Concordant     65.1    Somers' D    0.308
Percent Discordant     34.3    Gamma        0.310
Percent Tied            0.6    Tau-a        0.133
Pairs                  7670    c            0.654```

page 38 Table 2.3 Estimated coefficients for a multiple logistic regression model using the variables
lwt and race from the low birth weight study.

```proc logistic data=lowbwt1 descending covout outest=lowbwt2;
model low = lwt race2 race3;
run;
quit;The LOGISTIC Procedure                    Model Information
Data Set                      WORK.LOWBWT1
Response Variable             LOW                  {amp}lt; 2500g
Number of Response Levels     2
Number of Observations        189
Optimization Technique        Fisher's scoring         Response Profile Ordered                      Total
Value          LOW     Frequency
1            1            59
2            0           130                    Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.         Model Fit Statistics                              Intercept
Intercept         and
Criterion        Only        Covariates
AIC              236.672        231.259
SC               239.914        244.226
-2 Log L         234.672        223.259       Testing Global Null Hypothesis: BETA=0Test                 Chi-Square       DF     Pr {amp}gt; ChiSq
Likelihood Ratio        11.4129        3         0.0097
Score                   10.7572        3         0.0131
Wald                    10.1316        3         0.0175The LOGISTIC Procedure             Analysis of Maximum Likelihood Estimates                               Standard
Parameter    DF    Estimate       Error    Chi-Square    Pr {amp}gt; ChiSq
Intercept     1      0.8057      0.8452        0.9088        0.3404
LWT           1     -0.0152      0.00644       5.5886        0.0181
race2         1      1.0811      0.4881        4.9065        0.0268
race3         1      0.4806      0.3567        1.8156        0.1778           Odds Ratio Estimates             Point          95% Wald
Effect    Estimate      Confidence Limits
LWT          0.985       0.973       0.997
race2        2.948       1.133       7.672
race3        1.617       0.804       3.253Association of Predicted Probabilities and Observed Responses
Percent Concordant     64.1    Somers' D    0.293
Percent Discordant     34.8    Gamma        0.296
Percent Tied            1.1    Tau-a        0.127
Pairs                  7670    c            0.647```

page 42 Table 2.4 Estimated covariance matrix of the estimated coefficients in Table 2.3.

```proc print data=lowbwt2;
where _type_='COV';
var _name_ intercept lwt race2 race3;
run;Obs    _NAME_       Intercept            LWT      race2       race3 2     Intercept      0.71430    -.005213648     0.02260    -0.10350
3     LWT           -0.00521    0.000041465    -0.00065     0.00036
4     race2          0.02260    -.000647028     0.23819     0.05320
5     race3         -0.10350    0.000355854     0.05320     0.12722```

DescriptionUsageFormatSourceReferencesExamples

Data are from a hypothetical study to evaluate factors associated to low birth weight. This data set contains information
on 488 births to 188 women seen in an obstetrics clinic, comprising 1 to 3 births for each woman and yielding an average of 2.6 births per women.
The outcome binary variable «low» indicates low birth weight defined as birth weight less than 2500 grams. These data are
copyrighted by John Wiley {amp}amp; Sons Inc. and are used here to illustrate the
calculation of prevalence ratios and corresponding confidence intervals.

A data frame with 488 observations on the following 6 variables:

ID

Subject’s identification.

birth

Birth order.

smoke

Smoking Status During Pregnancy (1=Yes, 0= No).

race

Race (1=White, 0= Others).

age

Age of Mother (1= age greater than 35 years, 0= age less or equal to 35 years).

low

Low Birth Weight (1 if birth weight less than 2500 grams, 0 if birth weight greater or equal to 2500 grams).

The data come from
Data courtesy of University of Massachusetts Amherst
(http://www.umass.edu/statdata/statdata/stat-logistic.html)

These data are copyrighted by John Wiley {amp}amp; Sons Inc.
We acknowledged it and use the data to illustrate the methodology included in this library.

Hosmer and Lemeshow (2000) Applied Logistic Regression. New York: Wiley Inter-Science. Second Edition.

`Loadingrequiredpackage:bootLoadingrequiredpackage:stats4Loadingrequiredpackage:lme4Loadingrequiredpackage:MatrixLoadingrequiredpackage:HmiscLoadingrequiredpackage:latticeAttachingpackage:'lattice'Thefollowingobjectismaskedfrom'package:boot':melanomaLoadingrequiredpackage:survivalAttachingpackage:'survival'Thefollowingobjectismaskedfrom'package:boot':amlLoadingrequiredpackage:FormulaLoadingrequiredpackage:ggplot2Attachingpackage:'Hmisc'Thefollowingobjectsaremaskedfrom'package:base':format.pval,unitsIDbirthsmokeraceMin.:1.00Min.:1.000Min.:0.0000Min.:0.00001stQu.:46.751stQu.:1.0001stQu.:0.00001stQu.:0.0000Median:93.00Median:2.000Median:0.0000Median:0.0000Mean:93.56Mean:1.873Mean:0.3996Mean:0.14753rdQu.:139.253rdQu.:2.0003rdQu.:1.00003rdQu.:0.0000Max.:188.00Max.:4.000Max.:1.0000Max.:1.0000agelowMin.:0.00000Min.:0.00001stQu.:0.000001stQu.:0.0000Median:0.00000Median:0.0000Mean:0.06762Mean:0.30943rdQu.:0.000003rdQu.:1.0000Max.:1.00000Max.:1.0000`

## Compare maternal age to baby birth weight:

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## Description

The data come to us from Hosmer and Lemeshow (2000). Called the low
birth weight (lbw) data, the response is a binary variable, low,
which indicates whether the birth weight of a baby is under 2500g
(low=1), or over (low=0).

## Details

lbw is saved as a data frame.
Count models can use ftv as a response variable, or convert it to grouped format

## Examples

```data(lbw)
glmbwp {amp}lt;- glm(ftv ~ low   smoke   factor(race), family=poisson, data=lbw)
summary(glmbwp)
exp(coef(glmbwp))
library(MASS)
glmbwnb {amp}lt;- glm.nb(ftv ~ low   smoke   factor(race), data=lbw)
summary(glmbwnb)
exp(coef(glmbwnb))
```

### Basic Examples

#### Retrieve the ResourceObject:

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#### View the data:

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### Visualization

#### Compare maternal age to baby birth weight:

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## Format

A data frame with 189 observations on the following 10 variables.

`low`

1=low birthweight baby; 0=norml weight

`smoke`

1=history of mother smoking; 0=mother nonsmoker

`race`

categorical 1-3: 1=white; 2-=black; 3=other

`age`

age of mother: 14-45

`lwt`

weight (lbs) at last menstrual period: 80-250 lbs

`ptl`

number of false of premature labors: 0-3

`ht`

1=history of hypertension; 0 =no hypertension

`ui`

1=uterine irritability; 0 no irritability

`ftv`

number of physician visits in 1st trimester: 0-6

`bwt`

birth weight in grams: 709 — 4990 gr

## References

Hilbe, Joseph M (2007, 2011), Negative Binomial Regression, Cambridge University Press
Hilbe, Joseph M (2009), Logistic Regression Models, Chapman {amp}amp; Hall/CRC

## Retrieve the ResourceObject:

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## Source

Hosmer, D and S. Lemeshow (2000), Applied Logistic Regression, Wiley

## Usage

`data(lbw)`

## View the data:

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