Vidakovic B. Statistics for Bioengineering Sciences: With Matlab and WinBugs Support

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16.9 Exercises 651

(d) Discuss the adequacy of the proposed linear ﬁt. Predict the cholesterol

rate for lipids at the level 1.65 mg/kg/L

×10

−2

. Find the 95% conﬁdence and

prediction intervals.

16.10. Anscombe’s Data Sets. A celebrated classic example of the role of resid-

ual analysis and statistical graphics in statistical modeling was created by

Anscombe (1973). He constructed four quite different data sets (x

, y

), i =

1,... , 11 that share the descriptive statistics necessary to establish a linear

regression ﬁt

A linear model is appropriate for Data Set 1; the scatterplots and resid-

ual analysis suggest that Data Sets 2–4 seem not to be amenable to linear

modeling.

Set 1

10 8 13 9 11 14 6 4 12 7 5

8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68

Set 2

10 8 13 9 11 14 6 4 12 7 5

9.14 8.14 8.74 8.77 9.26 8.10 6.13 3.10 9.13 7.26 4.74

Set 3

10 8 13 9 11 14 6 4 12 7 5

7.46 6.77 12.74 7.11 7.81 8.84 6.08 5.39 8.15 6.42 5.73

Set 4

8 8 8 8 8 8 8 19 8 8 8

6.58 5.76 7.71 8.84 8.47 7.04 5.25 12.50 5.56 7.91 6.89

(a) Using MATLAB ﬁt the regression line for the four sets and provide four

ANOVA tables. What statistics are the same? What statistics differ?

(b) Plot the residuals against the ﬁtted values. Discuss the appropriateness

of the regressions.

16.11. Potato Leafhopper. The length of the developmental period (in days)

of the potato leafhopper, Empoasca fabae, from egg to adult seems to be

dependent on temperature (Kouskolekas and Decker, 1966). The original

data were weighted means, but for the purpose of this analysis we shall

consider them as though they were single observed values.

Temp

◦

F 59.8 67.6 70.0 70.4 74.0 75.3 78.0 80.4 81.4 83.2 88.4 91.4 92.5

Development (days) 58.1 27.3 26.8 26.3 19.1 19.0 16.5 15.9 14.8 14.2 14.4 14.6 15.3

(a) Find a 98% conﬁdence interval for unknown slope β

(b) Test the hypothesis that the intercept is equal to 60 against the alterna-

tive that it is larger than 60. Take

α =0.01.

ber of days) if the temperature is 85

◦

16.12. Cross-validating a Bayesian Regression. In this simulated exercise

covariates x

and x

are generated as

652 16 Regression

x1 = rand(1, 40); x2 = floor(10

rand(1,40)) + 1;

and the response variable y is obtained as

y = 2 + 6

x1 - 0.5

x2 + 0.8

randn(size(x1))

Write a WinBUGS program that selects 20 triples (x

, x

, y) to train the

linear regression model

= b

and then uses the remaining

20 triples to evaluate the model by comparing the original responses y

, i =

21,... , 40 with regression-predicted values

, i =21,... , 40. The comparison

involves calculating the MSE, the mean of (y

−

)

, i =21, . ..,40.

This is an example of how a cross-validation methodology is often employed

to assess statistical models.

How do the Bayesian estimators of

, β

, and σ compare to the “true”

values 2, 6,

−0.5, and 0.8?

16.13. Taste of Cheese. As cheddar cheese matures, a variety of chemical pro-

cesses take place. The taste of mature cheese is related to the concentration

of several chemicals in the ﬁnal product. In a study of cheddar cheese from

LaTrobe Valley of Victoria, Australia, samples of cheese were analyzed for

their chemical composition and were subjected to taste tests. The table be-

low presents data [from the experiments of G. T. Lloyd and E. H. Ramshaw,

CISRO Food Research, Victoria, Australia, analyzed in Moore and McCabe

(2006)] for one type of cheese manufacturing process. Taste is the response

variable of interest. The taste scores were obtained by combining scores

from several tasters. Three of the chemicals whose concentrations were

measured were acetic acid, hydrogen sulﬁde, and lactic acid. For acetic acid

and hydrogen sulﬁde log transformations were taken.

Taste Acetic H2S Lactic Taste Acetic H2S Lactic

12.3 4.54 3.13 0.86 20.9 5.16 5.04 1.53

39.0 5.37 5.44 1.57 47.9 5.76 7.59 1.81

5.6 4.66 3.81 0.99 25.9 5.70 7.60 1.09

37.3 5.89 8.73 1.29 21.9 6.08 7.97 1.78

18.1 4.90 3.85 1.29 21.0 5.24 4.17 1.58

34.9 5.74 6.14 1.68 57.2 6.45 7.91 1.90

0.7 4.48 3.00 1.06 25.9 5.24 4.94 1.30

54.9 6.15 6.75 1.52 40.9 6.37 9.59 1.74

15.9 4.79 3.91 1.16 6.4 5.41 4.70 1.49

18.0 5.25 6.17 1.63 38.9 5.44 9.06 1.99

14.0 4.56 4.95 1.15 15.2 5.30 5.22 1.33

32.0 5.46 9.24 1.44 56.7 5.86 10.20 2.01

16.8 5.37 3.66 1.31 11.6 6.04 3.22 1.46

26.5 6.46 6.92 1.72 0.7 5.33 3.91 1.25

13.4 5.80 6.69 1.08 5.5 6.18 4.79 1.25

(a) Find the equation in multivariable linear regression that predicts Taste

using Acetic, H2S, and Lactic as covariates.

16.9 Exercises 653

(b) For Acetic = 5, H2S = 8, and Lactic = 2 estimate the regression re-

sponse

and ﬁnd standard deviations for the mean and individual re-

sponses. [Ans. 43.37, 6.243, 11.886]

(d) Construct an ANOVA table.

(e) Find ordinary, studentized, and studentized deleted residuals for the

observation Y

(f) Find DFFITS

and COOKSD

(g) Find DFBETAS

on the Lactic coefﬁcient.

(h) Find and discuss the VIF.

16.14. Slowing the Progression of Arthritis. Arthritis is caused by the break-

down of collagen in joint cartilage by the enzyme MMP-13. The antibiotic

doxycycline is a general inhibitor of MMPs and, by inhibiting the activity

of MMP-13, is an effective method of slowing the progression of arthritis.

At present, doxycycline is used to treat both rheumatiod arthritis and os-

teoarthritis. The rabbit’s HIG-82 synovial cell line was used to model arthri-

tis. MMP-13 was prepared by adding PMA, which guarantees that its pres-

ence, and APMA, which activates it. The enzyme MMP-13 was mixed with

a quenched substrate. When the enzyme cleaves the substrate, it ﬂuoresces.

This ﬂuorescence was used to measure the amount of MMP-13 activity.

Doxycycline, which decreases the amount of enzyme activity, was added

in increasing concentrations: 0, 25, 50, 75, 100, and 200 micromols. The

decrease in the ﬂuorescence produced by the cleaved substrate when doxy-

cycline was present was used to measure the decrease in activity of the en-

zyme. The same experiment was performed on three different plates, and

the data was normalized.

The data set

arthritis1.mat can be found on the book’s Web page. The

data ﬁle contains 72 observations (rows); the ﬁrst column represents doxy-

cycline concentration (0, 25, 50, 75, 100 and 200), the second column is the

ﬂuorescence response, and the third column is the plate number.

(a) Fit the linear regression where ﬂuorescence is the response and doxycy-

cline concentration is the predictor. Predict the ﬂuorescence if the doxycy-

cline concentration is equal to 125.

(b) Since three plates are present, run

aoctool with doxycycline concentra-

tion as x, ﬂuorescence as y, and plate number as g. Is there a signiﬁcant

difference between the plates?

16.15. Insulin on Opossum Liver. Corkill (1932) provides data on the inﬂuence

of insulin on opossum liver. In the experimental setup the 20 animals (com-

mon gray Australian opossums – Trichosurus) fasted for 24 or 36 hours.

Ten animals, four from the 24-hour fasting group and six from the 36-hour

fasting group, were injected with insulin, while the remaining ten animals

served as controls, that is, they received no insulin. After 3 to 4 hours liver

glycogen and blood sugar were measured. The weights of the animals were

recorded as well.

654 16 Regression

The goal of the study was to explore the deposition of liver glycogen after the

insulin regimen in opossums. In rabbits and cats, for example, it was pre-

viously found that insulin induced signiﬁcant glycogen storage. This study

found a slight depletion of liver glycogen after the insulin treatment.

Our goal is to model the liver glycogen based on weight, level of blood sugar,

insulin indicator, and fasting regime.

Is the insulin indicator (0 no, 1 yes) an important covariate in the model?

Animal Weight Liver glycogen Blood sugar Fasting period Insulin

1 1502 1.80 0.124 24 0

2 1345 0.95 0.115 24 0

3 1425 1.12 0.128 24 0

4 1650 1.05 0.110 24 0

5 1520 0.45 0.052 24 1

6 1300 0.48 0.050 24 1

7 1250 0.75 0.045 24 1

8 1620 0.60 0.040 24 1

9 1725 0.76 0.130 36 0

10 1450 0.51 0.112 36 0

11 1800 0.48 0.105 36 0

12 1685 0.34 0.121 36 0

13 1560 0.38 0.116 36 0

14 1650 0.45 0.108 36 0

15 1650 0.65 0.032 36 1

16 1575 0.28 0.025 36 1

17 1260 0.10 0.045 36 1

18 1485 0.26 0.050 36 1

19 1520 0.18 0.030 36 1

20 1616 0.30 0.028 36 1

16.16. Slope in EIV Regression. Show that the EIV regression slope in (16.7)

tends to S

when η →0.

16.17. Interparticular Spacing and Wavelength in Nanoprisms 2. In the

context of Example 15.4, let x

= (separation)

−1

and y = log(wavelength).

The part of MATLAB regression output for

nonoprism.dat data set is

given below.

[b, bint, r, rint, stats] = regress(y,[ones(size(x)) x])

%b =

% -4.7182

% 1.6289

%bint =

% -5.6578 -3.7787

% 1.3061 1.9516

16.9 Exercises 655

%r =

% 0.0037

% -0.0084

% ...

% 0.0058

% 0.0046

%rint =

% -0.0101 0.0176

% -0.0232 0.0064

% ...

% -0.0096 0.0213

% -0.0110 0.0203

%stats =

% 0.8545 111.5735 0.0000 0.0001

Using information contained in this output, answer the following questions:

(a) What is the regression equation linking x and y?

(b) Predict y

=log(wavelength) for x =2.9. What is the wavelength for such

here and how is it interpreted? What is the F-statistic here?

Is it signiﬁcant?

(d) The 95% conﬁdence interval for the population slope

is [1.3061,

1.9516]. Using information in this output construct a 99% conﬁdence in-

terval for

MATLAB AND WINBUGS FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch16.Reg/

adhesions.m, ancova2.m, coburnreg.m, cpeptide.m, degradation.m,

diabetes.m, diagnostics.m, dissection.m, errorinvar.m, fatreg.m,

fatreg1.m, fatregdiag.m, galton.m, hemo.m, hubble.m, invitro.m,

kanamycin.m, kodlin.m, myeb.m, oldfaithful.m, pedometer1.m, ratwei.m,

tastecheese.m, vitalcapacity.m

ancovafibers.odc, fat.odc, mellitus.odc, regressionpred.odc,

vortex.odc

adhesion.xls, alcos.dat|xls, arthritis1.dat|mat, circalbumin.dat,

coburn.mat, Cpeptideext.dat|mat, fat.dat, fatreg1.m, galton.dat,

galtoncompact.dat, kanamycin.dat, Lactis.dat, nanoprism.dat,

pearson.dat, pmr1.mat, vitalcapacity.xlsx

656 16 Regression

CHAPTER REFERENCES

Anscombe, F. (1973). Graphs in statistical analysis. Am. Stat., 27, 17–21.

Armitage. P. and Berry, G. (1994). Statistical Methods in Medical Research. Blackwell Sci-

ence, London.

Carter, G. and Mitchell, C. (1958). Methods for adapting the virus of Rinderpest to rabbits.

Science, 128, 252–253.

Clark, R., Margraf, H., and Beauchamp, R. (1975). Fat and solid ﬁltration in clinical perfu-

sions. Surgery, 77, 216–224.

Corkill, B. (1932). The inﬂuence of insulin on the liver glycogen of the common grey aus-

tralian “opossum” (Trichosurus). J. Physiol., 75, 1, 29–32. PMCID: PMC1394507.

Galton, F. (1886). Regression towards mediocrity in hereditary stature. J. Anthropol. Inst.

Great Br. Ireland, 15, 246–263.

Grifﬁn, G. E., Abbott, W. E., Pride, M. P., Runtwyler, E., Mautz, F. R., and Grifﬁth, L. (1945).

Available (thiocyanate) volume total circulating plasma proteins in normal adults,

Ann. Surg., 121, 3, 352–360.

Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Chapman & Hall, London.

Kleinbaum, D. G., Kupper, L. L., and Muller, K. E. (1987). Applied Regression Analysis and

Other Multivariable Methods. PWS-Kent, Boston.

Kodlin, D. (1951). An application of the analysis of covariance in pharmacology. Arch. Int.

Pharmacodyn. Ther., 87, 1–2, 207–211.

Kouskolekas, C. and Decker, G. (1966). The effect of temperature on the rate of development

of the potato leafhopper, Empoasca fabae (Homoptera: Cicadelidae). Ann. Entomol.

Soc. Am., 59, 292–298.

Laud, P. and Ibrahim, J. (1995). Predictive model selection. J. R. Stat. Soc. Ser. B, 57, 1,

247–262.

Lopes, J. B., Dallan, L. A. O., Moreira, L. P. F., Carreiro, M. C., Rodrigues, F. L. B., Mendes,

P. C., and Stol, N. A. G. (2009). New quantitative variables to measure postoperative

pericardial adhesions. Useful tools in experimental research. Acta Cir. Bras., 24, São

Paulo.

Moore, D. and McCabe, G. (2006). Introduction to the Practice of Statistics, 5th edn. Free-

man, San Francisco.

Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS. Wiley, Hoboken.

Rawlings, J. O., Pantula, S., and Dickey, D. (1998). Applied Regression Analysis: A Research

Tool. Springer Texts in Statistics. Springer, Berlin Heidelberg New York.

Sockett, E. B., Daneman, D., Clarson, C., and Ehrich, R. M. (1987). Factors affecting and

patterns of residual insulin secretion during the ﬁrst year of type I (insulin dependent)

diabetes mellitus in children. Diabetes, 30, 453–459.

Yago, T., Lou, J., Wu, T., Yang, J., Miner, J. J., Coburn, L., Lopez, L. A., Cruz, M. A., Dong,

J.-F., McIntire, L. V., McEver, R. P., and Zhu, C. (2008). Platelet glycoprotein Iba forms

catch bonds with human WT vWF but not with type 2B von Willebrand disease vWF.

J. Clin. Invest., 118, 9, 3195–3207.

Chapter 17

Regression for Binary and Count Data

There are 10 types of people in the world, those who can read binary, and those who

can’t.

– Anonymous

WHAT IS COVERED IN THIS CHAPTER

• Logistic Regression: Fitting and Assessing the Model

• Probit and Complementary Log-Log Links

• Poisson Regression: Fitting and Assessing the Model

• Two Case Studies: Caesarean Sections and Danish IHGA Study

• Log-linear Models in Contingency Tables

17.1 Introduction

Traditional uni- or multivariable linear regression assumes a normally dis-

tributed response centered at a linear function of the predictors. For example,

in regression with a single predictor, the response y

is modeled as normal

N (β

+β

,σ

), where the expectation (conditional on covariate x

) is a lin-

ear function of x

For some regression scenarios this model is inadequate because the re-

sponse is not normally distributed. The response could be categorical, for ex-

ample, with two or more categories (“disease present – disease absent,” “sur-

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support,

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_17,

657

658 17 Regression for Binary and Count Data

vived – died,” “low – medium – high,” etc.) or be integer valued (“number with

the disease,” “number of failures,” etc.), and yet it may still depend on a covari-

ate or a vector of covariates, x. In this chapter we discuss logistic and Poisson

regressions that are appropriate for binary and counting responses.

In logistic regression the responses are binary, coded, without loss of gen-

erality, as 0 and 1 (or as 1 and 2 in WinBUGS). In Poisson regression the

responses are nonnegative integers well modeled by a Poisson distribution in

which the rate

λ depends on one or more covariates. The covariates enter the

model in a linear fashion; however, their connection with the (expected) re-

sponse is nonlinear.

Both logistic and Poisson regressions are examples of a wide class of models

called generalized linear models (GLMs). The term generalized linear model

refers to models introduced by Nelder and Wedderburn (1972) and popularized

by the monograph of McCullagh and Nelder (1982, second edition 1989). In a

canonical GLM model, the response variable y

is assumed to follow a distri-

bution from the class of distributions called the exponential family, with mean

µ, which is assumed to depend on covariates via their linear combination. The

exponential family is a rich family of distributions and includes almost all im-

portant distributions (normal, Bernoulli, binomial, Poisson, gamma, etc.). This

link between the mean

µ and covariates can be nonlinear, but the distribution

of y

depends on covariates only via their linear combination. Linear regres-

sion is a special case of GLMs for normally distributed responses, in which the

mean is directly modeled by a linear combination of covariates.

17.2 Logistic Regression

Assume that a response y

, depending on a covariate x

, is categorical and can

take two possible values. Examples of such responses include male–female,

sick–healthy, alive–dead, pass–fail, success–failure, win–loss, etc. One usually

assigns provisional numerical values to the responses, say 0 and 1, mainly to

simplify notation. Our interest is in modeling the probability of response y

given the observed covariates.

Classical least-squares regression y

= β

+β

+²

is clearly inadequate

since for unbounded x

the linear term β

+β

is unbounded as well. In

addition, the residuals have only two possible values, and the variance of y

not free of x

We assume that y is Bernoulli distributed with

E y = P(y = 1) = p. Since

E(y

|X = x

) = P(y

= 1|X = x

) = p

is a number between 0 and 1, it is reason-

able to model p

as F(β

+β

) for some probability CDF F. Equivalently,

−1

(

)

=β

+β

17.2 Logistic Regression 659

In principle, any monotone cumulative distribution function F can provide a

link between the probability p and the covariate(s), but the most used dis-

tributions are logistic, normal, and complementary log-log, leading to logistic,

probit, and clog-log regressions. The most popular among the three is the logis-

tic regression because its coefﬁcients, measuring the impact of the predictors

on the binary response y, have convenient interpretations via the log odds of

the events

{y =1}. In logistic regression, F

−1

(p) =log

1−p

is called the logit and

denoted as logit(p).

17.2.1 Fitting Logistic Regression

The basic statistical model for logistic regression is

∼ B er(p

), (17.1)

logit(p

) = log

1 − p

=β

+β

, i =1, . .., n

when the responses are Bernoulli, 0 or 1.

When multiple measurements correspond to the same covariate, it is con-

venient to express the responses as binomial counts, that is, n

responses cor-

responding to covariate x

are grouped together and y

is the number of re-

sponses equal to 1:

∼ B in(n

, p

), (17.2)

logit(p

) = log

1 − p

=β

+β

, i =1, . .., k,

i=1

= n.

In principle, it is always possible to express the binomial response model

via Bernoulli responses by repeating y

times the response 1 and n

−y

times

the response 0 for the same covariate x

. The converse is not possible in gen-

eral, especially if y

depends on a continuous covariate. Assessing the goodness

of ﬁt for models of type (17.1) is a well-known problem since the asymptotic

distributional results do not hold even for arbitrarily large sample sizes.

How does one estimate parameters

and β

in logistic model (17.1) or

(17.2)? The traditional least-squares algorithm that is utilized in linear re-

gression is not applicable. Estimating the model coefﬁcients amounts to solv-

660 17 Regression for Binary and Count Data

ing a nonlinear equation, and this is done by an iterative procedure. The al-

gorithm for a single predictor is illustrated and implemented in the m-ﬁle

logisticmle.m. There, the Newton–Raphson method is used to solve non-

linear likelihood equations and calculate coefﬁcients b

and b

as estimators of

population parameters

and β

. Details regarding the background and con-

vergence of methods for estimating model parameters are beyond the scope of

this text and can be found in McCullagh and Nelder (1989).

Once the parameters

and β

are estimated, the probability p

= P(y =

1|x = x

) is obtained as

exp{b

}

1 +exp{b

}

1 +exp{−b

−b

}

In addition to the nature of response y

, there is another key difference

between ordinary and logistic regressions. For linear regression the variance

does not depend on the mean

+β

; it is constant for all x

. This is one

of the assumptions for linear regression. In logistic regression the variance is

not constant; it is a function of the mean. From (17.1),

E y

= p

and Var y

(1 −E y

If p

−1 covariates (p ≥ 2 parameters) are available, as is often the case,

then

b =`

= b

+···+b

p−1

i,p−1

replaces b

, where X

is the ith row of a design matrix X of size n × p,

and b

= (b

... b

p−1

)

. Now,

exp{X

1 +exp{X

1 +exp{−X

with b maximizing the log-likelihood

log L(

β) =`(β) =

i=1

·(X

β) −

i=1

log

1 +exp{X

β}

Example 17.1. Caesarean-Section Infections. A Caesarean-section, or C-

section, is major abdominal surgery, so moms who undergo C-sections are

more likely to have an infection, excessive bleeding, blood clots, more post-

partum pain, a longer hospital stay, and a signiﬁcantly longer recovery. The