Generalized Linear Models - Research-Training.net · Generalized Linear Model (GLM); a statistical...

Generalized Linear Models

Methods@Manchester Summer SchoolManchester University

July 2–6, 2018

Generalized Linear Models: a generic approach to

statistical modelling

www.research-training.net/Manchester2018

[email protected]

University of Manchester

Graeme Hutcheson Methods@Manchester Summer School, 2018

The slides and R-files for this session are available on the coursewebsite...

www.research-training.net/Manchester2018

Lecture Slides:

http://www.research-training.net/addedfiles/

2018Manchester/2018Manchester04GLM.pdf

R-notebook:

http://www.research-training.net/addedfiles/

2018Manchester/2018Manchester04.Rmd

...or from the course DVD.

This course uses a system of analysis that represents researchquestions in the form of equations. For example...

mathematics test score ∼ gender

success (yes/no) ∼ age

salary ∼ gender + age + ethnicity

number of arrests ∼ age*gender

Representing research questions in this way explicitly identifies therelationships to be tested, the structure of the data and how themodel is entered into the analysis programme.

The formulation of the research question in equation format is alsouseful as it is the same representation as that used by theGeneralized Linear Model (GLM); a statistical model that may beapplied to a range of analytical problems.

This lecture provides a relatively non-technical introduction to theGLM. Those looking for more detailed treatments are advised tolook at the following texts...

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models(2nd edition). Chapman & Hall/CRC.

Hutcheson, G. D. and Sofroniou, N. (1999). The MultivariateSocial Scientist: Introductory statistics using generalizedlinear models. Sage Publications.

The formulation of the research question in equation format is alsouseful as it is the same representation as that used by theGeneralized Linear Model (GLM); a statistical model that may beapplied to a range of analytical problems.

This lecture provides a relatively non-technical introduction to theGLM. Those looking for more detailed treatments are advised tolook at the following texts...

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models(2nd edition). Chapman & Hall/CRC.

Hutcheson, G. D. and Sofroniou, N. (1999). The MultivariateSocial Scientist: Introductory statistics using generalizedlinear models. Sage Publications.

The Generalized Linear Model

In it’s simplest form, the GLM is a statistical technique thatpredicts a single variable (the response variable), using one or moreother variables (the explanatory variables).

The response and explanatory variables (also known as the randomand systematic components of the model) are linked (∼) accordingto a function that takes account of the measurement scale of theresponse variable.

response variable ∼ explanatory variables

There are many link functions that are available for GLM modelsto take account of the different ways in which the randomcomponent (the variable that is being predicted) is distributed (eg.as a number, category, count, skewed, etc.). This courseintroduces three links that enable continuous, categorical andcount response variables to be modelled.

To model a continuous response variable, an identity link is used.To model a count response variable, a log link is used. To model acategorical variable, a logit link is used.

response research link linearvariable equation function model

continuous Y ∼ X identity Y = X

count Y ∼ X log log(Y ) = X

categorical Y ∼ X logit logit(Y ) = X

These will be explained in detail in later sessions. It is enough atthis point to just realise that different response variables can bemodelled by changing the link function.

In practice, if we know the measurement scale of the variable beingpredicted, we can identify an appropriate GLM technique to use...

I If Y is continuous: OLS regression.

I If Y is a count: Poisson regression.

I If Y is ordered categorical: Proportional-odds regression.

I If Y is unordered categorical: Multinomial regression.

GLM models are particularly powerful, as they are all conceptuallyvery similar. Learning to apply and interpret results from onetechnique greatly helps in applying and interpreting results fromthe others.

GLM — a statistical representation

Up till now models have been represented using variable names.This way of looking at models is very useful as it is the way thatmodels are conceptualised and input into the statistical software.

It is useful, however, to also represent models using a moredetailed statistical representation; one that corresponds to the waythat the results are produced and reported.

The statistical representation simply includes some parameters thatquantify the relationships in the data. We not only want to knowthat Y and X are related, we also want to know HOW they arerelated. ie. When X changes by a set amount, what is the effecton Y ?

GLM — the parameters

A conceptual model of ice cream consumption (from the IceCream

dataset) is...

consumption ∼ temperature

which is represented statistically as...

consumption = β0 + β1 temperature

β0 estimates ‘consumption’ when ‘temperature’ is zero.

β1 estimates the change in ‘consumption’ for a unit increase in‘temperature’.

dataset) is...

The most interesting statistic for us is the parameter that estimatesthe relationship between ‘temperature’ and ‘consumption’; theparameter β1. The formal description of this parameter is...

For a unit increase in X , the estimated change in Y is β1.

For a unit increase in ‘temperature’, the estimated change in‘consumption’ is β1.

The values for the parameters are obtained from the statisticaloutput for the model...

β0 = 0.207 and β1 = 0.003.

The following output was obtained by putting the model‘consumption ∼ temperature’ into the Rcmdr GLM menu (asconsumption is continuous, select the identity link from theGaussian family)...

glm(formula = Consumption ~ Temperature,

family = gaussian(identity),

data = IceCream)

Coefficients:

Estimate Std. Error

(Intercept) 0.2068621 0.0247002

Temperature 0.0031074 0.0004779

These parameters are also represented in the effect display... (usethe Rcmdr menu options ‘Models, graphs, effect plot’....).

The following output was obtained by putting the model‘consumption ∼ temperature’ into the Rcmdr GLM menu (asconsumption is continuous, select the identity link from theGaussian family)...

data = IceCream)

Coefficients:

Estimate Std. Error

(Intercept) 0.2068621 0.0247002

Temperature 0.0031074 0.0004779

These parameters are also represented in the effect display... (usethe Rcmdr menu options ‘Models, graphs, effect plot’....).

Temperature effect plot

Temperature

0 10 20 30 40 50 60 70

computing β1

As temperature increases by40 (30 to 70), consumptionincreases by 0.125 (0.3 to0.425). For a unit increase intemperature, consumptionincreases by 0.003 ( 0.125

computing β0

The value of ‘consumption’when ‘temperature’ = 0. Asimple estimation from thegraph shows β0 = 0.2.

Temperature

0 10 20 30 40 50 60 70

computing β1

computing β0

Temperature

0 10 20 30 40 50 60 70

computing β1

computing β0

Categorical explanatory variables...

It is useful at this stage to look at categorical explanatoryvariables. A detailed description of analysing categoricalexplanatory variables is available in...

Hutcheson, G. D. (2011). Categorical Explanatory Variables.Journal of Modelling in Management. 6, 2: 225–236. http://www.

research-training.net/READING/JMM6(2)contrastCODINGupdated.pdf

At a basic level, categorical variables are divided into a number ofbinary comparisons. Each category is then compared to a specificcategory (the reference).

The following slide shows a model of examination score andwhether this is related to the school a child is at. school is anunordered categorical variable with three categories (schoolA,schoolB and schoolC).

The conceptual model

examination score ∼ school

is represented statistically as

score = β0 + β1 schoolB + β2 schoolC

β0 estimates ‘score’ when ‘schoolB’ and ‘schoolC’ are both zero(this is equivalent to the score for schoolA).

β1 estimates the change in ‘score’ for schoolB compared toschoolA.

β2 estimates the change in ‘score’ for schoolC compared toschoolA.

The following output from the ‘schools.csv’ dataset wasobtained by putting the model ‘score ∼ school’ into the RcmdrGLM menu (as consumption is continuous, identify the identitylink)...

glm(formula = SCORE ~ SCHOOL,

data = schools)

Coefficients:

Estimate Std. Error

(Intercept) 63.200 3.058

SCHOOL[T.schoolB] 7.400 4.325

SCHOOL[T.schoolC] -7.600 4.325

The predicted score at schoolA is 63.2. SchoolB is 7.4 pointshigher (70.6) and schoolC is 7.6 points lower (56.6).

The following output from the ‘schools.csv’ dataset wasobtained by putting the model ‘score ∼ school’ into the RcmdrGLM menu (as consumption is continuous, identify the identitylink)...

glm(formula = SCORE ~ SCHOOL,

data = schools)

Coefficients:

Estimate Std. Error

(Intercept) 63.200 3.058

SCHOOL[T.schoolB] 7.400 4.325

SCHOOL[T.schoolC] -7.600 4.325

The predicted score at schoolA is 63.2. SchoolB is 7.4 pointshigher (70.6) and schoolC is 7.6 points lower (56.6).

This result can be easily seen in the effect display thataccompanies the model...

SCHOOL

schoolA schoolB schoolC

β0 = 63.2

The value of ‘score’ for‘schoolA’ (the referencecategory).

β1 = 70.6

schoolB is 7.4 units higher thanschoolA.

β2 = 56.6

schoolC is 7.6 units lower thanschoolA.

SCHOOL

β0 = 63.2

β1 = 70.6

β2 = 56.6

SCHOOL

β0 = 63.2

β1 = 70.6

β2 = 56.6

SCHOOL

β0 = 63.2

β1 = 70.6

β2 = 56.6

interpreting parameters for other GLMs

The interpretation of the regression coefficients is essentially thesame for all GLMs. For example, in order to model the number of‘checks’ a suspect appears on and whether this is related to aperson’s age (these data are from the ‘Arrests’ dataset from theeffects package)...

number of checks ∼ age

as the number of checks is a count variable, a log-link should beused. The statistical model for this is...

log (checks) = β0 + β1 age

The parameter β0 indicates the value of log(checks) when ‘age’equals zero. The parameter β1 indicates the change in log(checks)for a unit change in ‘age’.

The following output was obtained by putting the model ‘checks ∼age’ into the Rcmdr GLM menu (as checks is count, identify thelog link from the Poisson family)...

glm(formula = checks ~ age,

family = poisson(log),

data = Arrests)

Coefficients:

Estimate Std. Error

(Intercept) 0.150803 0.031680

age 0.014028 0.001196

When ‘age’ = 0, ‘log(count)’ = 0.151.

For a unit increase in ‘age’, ‘log(count)’ changes by 0.014.

The following output was obtained by putting the model ‘checks ∼age’ into the Rcmdr GLM menu (as checks is count, identify thelog link from the Poisson family)...

glm(formula = checks ~ age,

family = poisson(log),

data = Arrests)

Coefficients:

Estimate Std. Error

(Intercept) 0.150803 0.031680

age 0.014028 0.001196

When ‘age’ = 0, ‘log(count)’ = 0.151.

For a unit increase in ‘age’, ‘log(count)’ changes by 0.014.

This result can be easily seen in the effect display thataccompanies the model... The follow graph shows the relationshipbetween age and log(count)...

0 10 20 30 40 50 60

β0 = 0.151 (see above)

A simple estimation from thegraph shows that when‘age’=0, log(checks)=0.15.

As ‘age’ increases from 20 to60 (increase of 40), log(checks)increases from 0.43 to 1(increase of 0.57). For eachunit increase in ‘age’,log(checks) increases by 0.57

0.014.

0 10 20 30 40 50 60

0.014.

0 10 20 30 40 50 60

0.014.

The model parameters for GLM models have an easyinterpretation; one that is essentially the same for all GLM models.

The parameters can be interpreted from the ‘standard’ statisticaloutput shown in the tables, or from the effect displays.

The model parameters for GLM models have an easyinterpretation; one that is essentially the same for all GLM models.

The parameters can be interpreted from the ‘standard’ statisticaloutput shown in the tables, or from the effect displays.

Assessing significance...

Assessing significance for GLMs

In order to interpret the models, it is useful to know thesignificance associated with each of the parameter estimates.Although ‘consumption’ may increase by 0.003 for each unitincrease in ‘temperature’; there is no indication whether thisincrease may have been due to chance.

In addition to the parameter estimates, we usually also want toknow which parameters, or groups of parameters are significant.

This is where the GLM models excel, as they employ a simplemethod for determining significance; one that applies generally toall GLMs.

Deviance

Significance is assessed in GLMs using a common method based ona statistic known as the deviance.

The deviance is simply a measure of the difference between thevalues predicted by the model and the actual values (predictedvalues compared to the observed data). If the model provides agood prediction of the response variable, the deviance will berelatively small. If the model does not provide accurate predictionsof the response variable, the deviance will be relatively large.

The deviance basically gives an indication of how well the modelfits the data.

Deviance

Using the deviance, it is easy to determine the significance ofindividual and/or groups of variables by comparing nested models.For example, if we wanted to assess whether temperature is asignificant predictor of ice cream consumption, we can compare amodel of consumption that includes temperature, with a modelthat does not.

consumption = β0 + β1 temperature deviance = 0.050

consumption = β0 deviance = 0.126

The ‘effect’ that temperature has had on the model is to reducethe deviance by 0.076 (0.126 - 0.050). All of this information isgiven in the analysis output...

The Null deviance is the deviance in the response variable withouttaking into account any other information. The Residual devianceis the deviance in the model that includes the explanatory variables.

family = gaussian(identity), data = IceCream)

Coefficients:

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

(Intercept) 0.2068621 0.0247002 8.375 4.13e-09

Temperature 0.0031074 0.0004779 6.502 4.79e-07

Null deviance: 0.125523 on 29 degrees of freedom

Residual deviance: 0.050009 on 28 degrees of freedom

Analysis of Deviance Table (Type II tests)

Response: Consumption

SS Df F Pr(>F)

Temperature 0.075514 1 42.28 4.789e-07

Residuals 0.050009 28

The model above was obtained by running a GLM on the IceCreamdata and then requesting an analysis of deviance table. Thesecommands can be issued via the Rcmdr menus, or direct to ascript file using the commands...

GLM.1 <- glm(Consumption ~ Temperature,

family=gaussian(identity),

data=IceCream)

summary(GLM.1)

Anova(GLM.1, type="II", test="F")

The change in deviance associated with removing ‘temperature’from the model is assessed for significance using the “F-test”. Adetailed description of significance tests and deviance is providedin...

Hutcheson, G. D. and Moutinho, L. (2008). Statistical Modellingfor Management. Sage Publications.

The model above was obtained by running a GLM on the IceCreamdata and then requesting an analysis of deviance table. Thesecommands can be issued via the Rcmdr menus, or direct to ascript file using the commands...

GLM.1 <- glm(Consumption ~ Temperature,

family=gaussian(identity),

data=IceCream)

summary(GLM.1)

Anova(GLM.1, type="II", test="F")

The change in deviance associated with removing ‘temperature’from the model is assessed for significance using the “F-test”. Adetailed description of significance tests and deviance is providedin...

Hutcheson, G. D. and Moutinho, L. (2008). Statistical Modellingfor Management. Sage Publications.

Temperature

30 35 40 45 50 55 60 65 70

significance oftemperature

a visual indication of thesignificance oftemperature can be seenin the effect display.

It is easy to see from theline and associatedconfidence intervals shownin the graph (the shadedarea of the plot) thatpredictions of consumptionare different astemperature changes.

Information abouttemperature is, therefore,important for predictingconsumption.

Temperature

30 35 40 45 50 55 60 65 70

significance oftemperature

a visual indication of thesignificance oftemperature can be seenin the effect display.

It is easy to see from theline and associatedconfidence intervals shownin the graph (the shadedarea of the plot) thatpredictions of consumptionare different astemperature changes.

Information abouttemperature is, therefore,important for predictingconsumption.

The significance of temperature in the model can be manuallycalculated by comparing the nested models...

model01: consumption = β0

model02: consumption = β0 + β1 temperature

This can be easily achieved using the Rcmdr Models, Hypothesistests, Compare two models... menu option.

The output clearly shows which models are being compared. Notethat the statistics for Temperature are the same as those providedpreviously...

Rcmdr> anova(model01, model02, test="F")

Analysis of Deviance Table

Model 1: Consumption ~ 1

Model 2: Consumption ~ Temperature

Resid. Df Resid. Dev Df Deviance F Pr(>F)

1 29 0.125523

2 28 0.050009 1 0.075514 42.28 0.0000004789

The same underlying theory may be applied to testing categoricalexplanatory variables and different response variables. Thefollowing example shows a binary response variable being predictedby a categorical and a continuous explanatory variable. These dataare from the Arrests dataset in the ‘effects’ package (toreproduce these results, don’t forget to change the variable ‘year’to a categorical variable ‘yearCAT’).

The model we will be investigating is...

released ∼ yearCAT + age

logit(released) = β0 + β1 year1998 + β2 year1999 +

β3 year2000 + β4 year2001 +

β5 year2002 + β6 age

β3 year2000 + β4 year2001 +

or represented more succinctly as...

logit(released) = β0 +1∑

βy(1−5)yearCAT + β6 age

The categorical variable ‘yearCAT’ is represented using 5 binarycomparisons; each year is compared to the reference category.

A GLM (using the logit link) is shown below...

glm(formula = released ~ age + yearCAT,

family = binomial(logit), data = Arrests)

Coefficients:

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

(Intercept) 1.48937 0.14841 10.036 < 2e-16

age -0.01294 0.00423 -3.059 0.00222

yearCAT[T.1998] 0.36712 0.13857 2.649 0.00806

yearCAT[T.1999] 0.43182 0.13360 3.232 0.00123

yearCAT[T.2000] 0.63258 0.13362 4.734 2.2e-06

yearCAT[T.2001] 0.41696 0.13112 3.180 0.00147

yearCAT[T.2002] 0.21149 0.18451 1.146 0.25172

Null deviance: 4776.3 on 5225 degrees of freedom

Residual deviance: 4743.2 on 5219 degrees of freedom

The logit model of ‘released’ shows the deviance scores at thebottom of the tabular output. The Null deviance tells us howmuch deviance there is in the variable ‘released’ (an “empty”model) and the Residual deviance tells us how much deviancethere is in the model with all the explanatory variables included.

In order to find out if each variable is significant, we need to lookat the ‘analysis of deviance table’...

Response: released

LR Chisq Df Pr(>Chisq)

age 9.0948 1 0.0025634 **

yearCAT 23.7183 5 0.0002459 ***

The logit model of ‘released’ shows the deviance scores at thebottom of the tabular output. The Null deviance tells us howmuch deviance there is in the variable ‘released’ (an “empty”model) and the Residual deviance tells us how much deviancethere is in the model with all the explanatory variables included.

In order to find out if each variable is significant, we need to lookat the ‘analysis of deviance table’...

Response: released

LR Chisq Df Pr(>Chisq)

age 9.0948 1 0.0025634 **

yearCAT 23.7183 5 0.0002459 ***

From the Analysis of Deviance table, we can see that includingyearCAT in the model results in a reduction in deviance of23.7183. This is the result of comparing the deviance of thefollowing models...

β3 year2000 + β4 year2001 +

logit(released) = β0 + β6 age

This change in deviance is significant and provides evidence thatthe variable yearCAT may be influential in predicting whether ornot someone is released.

From the Analysis of Deviance table, we can see that includingyearCAT in the model results in a reduction in deviance of23.7183. This is the result of comparing the deviance of thefollowing models...

β3 year2000 + β4 year2001 +

logit(released) = β0 + β6 age

This change in deviance is significant and provides evidence thatthe variable yearCAT may be influential in predicting whether ornot someone is released.

These models can be compared using the Rcmdr Models,Hypothesis tests, Compare two models... menu option.

Model 1: released ~ age

Model 2: released ~ yearCAT + age

Resid. Df Resid. Dev Df Deviance Pr(>Chi)

1 5224 4766.9

2 5219 4743.2 5 23.718 0.0002459

Including the variable age in the model results in a reduction indeviance of 9.0948. This is the result of removing 1 parameterfrom the model (df = 1).

β3 year2000 + β4 year2001 +

β5 year2002

This change in deviance is significant and provides evidence thatthe variable age may be influential in predicting whether or notsomeone is released.

Including the variable age in the model results in a reduction indeviance of 9.0948. This is the result of removing 1 parameterfrom the model (df = 1).

β3 year2000 + β4 year2001 +

β5 year2002

This change in deviance is significant and provides evidence thatthe variable age may be influential in predicting whether or notsomeone is released.

These models can be compared using the Rcmdr Models,Hypothesis tests, Compare two models... menu option.

Model 1: released ~ yearCAT

Model 2: released ~ yearCAT + age

Resid. Df Resid. Dev Df Deviance Pr(>Chi)

1 5220 4752.3

2 5219 4743.2 1 9.0948 0.002563

These results can also be inferred from the effect displays.Although these do not provide estimates of significance, thegraphics provide enough information to identify the importantrelationships visually.

yearCAT effect plot

yearCAT

199719981999200020012002

●●

age effect plot

20 25 30 35 40 45 50 55 60

The significance of each category to the prediction of the responsevariable in logit models is estimated using the z-distribution in the‘standard’ regression output. It should be noted that this is alarge-sample approximation and the deviance statistic is preferablefor assessing significance. Full information about this is availablein...

Hutcheson, G. D. and Moutinho, L. (2008). Statistical Modelling forManagement. Sage Publications.

Type II and Type III ANOVA tests

The choice of ANOVA test that is used to compare models is veryimportant, particularly for models with interactions.

Type III ANOVA tests are those that are computed for individualparameters and are provided in the ‘standard’ tabular output.

Type II ANOVA tests are those that are computed for variables andare provided in the ‘Analysis of Deviance’ table.

The difference between the two types of test is very important andwill be highlighted using examples later in the course. Thedifference between the two can be illustrated using the followingmodel (from the ICEcream data) which contains an interaction...

consumption ∼ temperature ∗ income

Type III tests

Coefficients:

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

(Intercept) 1.216e-01 3.023e-01 0.402 0.691

Temperature -1.687e-03 6.299e-03 -0.268 0.791

Income 7.459e-04 3.546e-03 0.210 0.835

Temperature:Income 6.250e-05 7.508e-05 0.832 0.413

Type II tests

SS Df F Pr(>F)

Temperature 0.087836 1 62.6925 2.145e-08 ***

Income 0.012611 1 9.0008 0.005883 **

Temperature:Income 0.000971 1 0.6929 0.412755

The significance of the variables temperature and Income aredifferent depending on the type of test used.

The type III test for temperature compares the models...

consumption = β0 + β1 temperature + β2 income + β3 temperature:income

consumption = β0 + β2 income + β3 temperature:income

whilst the type II test for temperature compares the models...

consumption = β0 + β1 temperature + β2 income

consumption = β0 + β2 income

The significance of the variables temperature and Income aredifferent depending on the type of test used.

The type III test for temperature compares the models...

consumption = β0 + β1 temperature + β2 income + β3 temperature:income

consumption = β0 + β2 income + β3 temperature:income

whilst the type II test for temperature compares the models...

consumption = β0 + β1 temperature + β2 income

consumption = β0 + β2 income

The type III test does not provide an indication of the significanceof the variable temperature. We need to compare a model thatincludes temperature with a model thatdoes not - BOTH of thesemodesl include temperature.

The type II test does compare models differentiated on the basis oftemperature and are, therefore, more appropriate for assessing theeffect of temperature.

The type III test does not provide an indication of the significanceof the variable temperature. We need to compare a model thatincludes temperature with a model thatdoes not - BOTH of thesemodesl include temperature.

The type II test does compare models differentiated on the basis oftemperature and are, therefore, more appropriate for assessing theeffect of temperature.

Traditional tests andequivalent GLM models...

Traditional tests and equivalent GLM models

The GLM models reproduce or replace many of the traditionaltests. For example, tests for independent group designs...

Traditional Test GLM

one independent variable

t-test (unrelated)

Mann-Whitney

1-way ANOVA (unrelated) Y ∼ X

Kruskal-Wallis

Jonck-heere Trend

chi-square (contingency table)

etc., etc.

multiple independent variables

complex selection of multi-way ANOVA models

multi-way contingency tables (log-linear) Y ∼ X1 + X2

The GLM models reproduce or replace many of the traditionaltests. For example, tests for independent group designs...

t-test (unrelated)

Mann-Whitney

1-way ANOVA (unrelated) Y ∼ X

Kruskal-Wallis

Jonck-heere Trend

chi-square (contingency table)

etc., etc.

multi-way contingency tables (log-linear) Y ∼ X1 + X2

... and tests for dependent (or matched) group designs...

paired t-test

Wilcoxon

1-way ANOVA (related) Y ∼ subject + X

Friedman

Pages L-trend

etc., etc.,

multi-way contingency tables (log-linear) Y ∼ subject + X1 + X2

In order to realise the power of the GLMs, it is important tounderstand the equivalence of the ‘traditional’ tests and the GLMs.Detailed information about this is provided in...

Hutcheson, G. D. and Schaefer, L. (2012). Test selection in the 21st century.Journal of Modelling in Management, 7,3: 375–387. http:

//www.research-training.net/READING/JMM7(3)testSELECTION.pdf.

In order to realise the power of the GLMs, it is important tounderstand the equivalence of the ‘traditional’ tests and the GLMs.Detailed information about this is provided in...

Hutcheson, G. D. and Schaefer, L. (2012). Test selection in the 21st century.Journal of Modelling in Management, 7,3: 375–387. http:

//www.research-training.net/READING/JMM7(3)testSELECTION.pdf.

The usefulness of the equation format for representing statisticalmodels is evident when the analyses are run in the software.

In order to select the appropriate GLM...

1. define equation

2. Identify scale of measurement for the response variable

3. Select the appropriate model in the Rcmdr.

I Generalized linear model... for continuous and count responsesI Multinomial logit model... for unordered categorical responsesI Ordinal regression model... for ordered categorical responses

1. define equation

3. Select the appropriate model in the Rcmdr.I Generalized linear model... for continuous and count responses

I Multinomial logit model... for unordered categorical responsesI Ordinal regression model... for ordered categorical responses

1. define equation

3. Select the appropriate model in the Rcmdr.I Generalized linear model... for continuous and count responsesI Multinomial logit model... for unordered categorical responses

I Ordinal regression model... for ordered categorical responses

1. define equation

3. Select the appropriate model in the Rcmdr.I Generalized linear model... for continuous and count responsesI Multinomial logit model... for unordered categorical responsesI Ordinal regression model... for ordered categorical responses

Then enter the equation...

Exercises...

Your research model is...

salary ∼ age + qualification

where salary and age are continuous and qualification is recordedin 4 ordered categories.

1. which link function is appropriate for this model?

2. How many parameters would you expect to see in the modelfor qualification?

3. which statistical technique would you use?

4. what is the statistical model (the linear model)?

A-level ∼ IQ + gender

where A-level is recorded as 6 ordered categories, IQ is recordedas continuous, and gender is recorded as 2 unordered categories.

2. How many parameters would you expect to see in the modelfor IQ?

traffic violations ∼ age ∗ gender

where traffic violations is recorded as the number ofrecorded violations (0 to 7), age is recorded as continuous, andgender is recorded as 2 unordered categories.

2. How many parameters would you expect to see in the model(including the intercept)?

5. Would you use a TYPE II or a TYPE III test to ascertain ifage is significant?

Holiday destination ∼ age ∗ gender

where Holiday destination is recorded as 6 unorderedcategories, age is recorded as continuous, and gender is recordedas 2 unordered categories.

3. Would you use a TYPE II or a TYPE III test to ascertain ifgender is significant?

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