Generalized Linear Models

Transcription

1 Generalized Linear Models Summer School Manchester University July 2 6, 2018 Generalized Linear Models: a generic approach to statistical modelling Graeme.Hutcheson@manchester.ac.uk University of Manchester

2 The slides and R-files for this session are available on the course website... Lecture Slides: Manchester/2018Manchester04GLM.pdf R-notebook: Manchester/2018Manchester04.Rmd...or from the course DVD.

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

4 This course uses a system of analysis that represents research questions in the form of equations. For example... mathematics test score gender

5 This course uses a system of analysis that represents research questions in the form of equations. For example... mathematics test score gender success (yes/no) age

6 This course uses a system of analysis that represents research questions in the form of equations. For example... mathematics test score gender success (yes/no) age salary gender + age + ethnicity

7 This course uses a system of analysis that represents research questions in the form of equations. For example... mathematics test score gender success (yes/no) age salary gender + age + ethnicity number of arrests age*gender

8 This course uses a system of analysis that represents research questions 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 the relationships to be tested, the structure of the data and how the model is entered into the analysis programme.

9 The formulation of the research question in equation format is also useful as it is the same representation as that used by the Generalized Linear Model (GLM); a statistical model that may be applied to a range of analytical problems.

10 The formulation of the research question in equation format is also useful as it is the same representation as that used by the Generalized Linear Model (GLM); a statistical model that may be applied to a range of analytical problems. This lecture provides a relatively non-technical introduction to the GLM. Those looking for more detailed treatments are advised to look 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 Multivariate Social Scientist: Introductory statistics using generalized linear models. Sage Publications.

11 The Generalized Linear Model In it s simplest form, the GLM is a statistical technique that predicts a single variable (the response variable), using one or more other variables (the explanatory variables).

12 The Generalized Linear Model In it s simplest form, the GLM is a statistical technique that predicts a single variable (the response variable), using one or more other variables (the explanatory variables). The response and explanatory variables (also known as the random and systematic components of the model) are linked ( ) according to a function that takes account of the measurement scale of the response variable.

13 The Generalized Linear Model In it s simplest form, the GLM is a statistical technique that predicts a single variable (the response variable), using one or more other variables (the explanatory variables). The response and explanatory variables (also known as the random and systematic components of the model) are linked ( ) according to a function that takes account of the measurement scale of the response variable. response variable explanatory variables

14 The Generalized Linear Model In it s simplest form, the GLM is a statistical technique that predicts a single variable (the response variable), using one or more other variables (the explanatory variables). The response and explanatory variables (also known as the random and systematic components of the model) are linked ( ) according to a function that takes account of the measurement scale of the response variable. response variable explanatory variables There are many link functions that are available for GLM models to take account of the different ways in which the random component (the variable that is being predicted) is distributed (eg. as a number, category, count, skewed, etc.). This course introduces three links that enable continuous, categorical and count response variables to be modelled.

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

16 To model a continuous response variable, an identity link is used. To model a count response variable, a log link is used. To model a categorical variable, a logit link is used. response research link linear variable equation function model continuous Y X identity Y = X count Y X log log(y ) = X categorical Y X logit logit(y ) = X

17 To model a continuous response variable, an identity link is used. To model a count response variable, a log link is used. To model a categorical variable, a logit link is used. response research link linear variable 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 at this point to just realise that different response variables can be modelled by changing the link function.

18 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression.

19 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression. If Y is a count: Poisson regression.

20 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression. If Y is a count: Poisson regression. If Y is ordered categorical: Proportional-odds regression.

21 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression. If Y is a count: Poisson regression. If Y is ordered categorical: Proportional-odds regression. If Y is unordered categorical: Multinomial regression.

22 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression. If Y is a count: Poisson regression. If Y is ordered categorical: Proportional-odds regression. If Y is unordered categorical: Multinomial regression.

23 In practice, if we know the measurement scale of the variable being predicted, we can identify an appropriate GLM technique to use... If Y is continuous: OLS regression. If Y is a count: Poisson regression. If Y is ordered categorical: Proportional-odds regression. If Y is unordered categorical: Multinomial regression. GLM models are particularly powerful, as they are all conceptually very similar. Learning to apply and interpret results from one technique greatly helps in applying and interpreting results from the others.

24 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 that models are conceptualised and input into the statistical software.

25 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 that models are conceptualised and input into the statistical software. It is useful, however, to also represent models using a more detailed statistical representation; one that corresponds to the way that the results are produced and reported.

26 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 that models are conceptualised and input into the statistical software. It is useful, however, to also represent models using a more detailed statistical representation; one that corresponds to the way that the results are produced and reported. The statistical representation simply includes some parameters that quantify the relationships in the data. We not only want to know that Y and X are related, we also want to know HOW they are related. ie. When X changes by a set amount, what is the effect on Y?

27 GLM the parameters A conceptual model of ice cream consumption (from the IceCream dataset) is... consumption temperature

28 GLM the parameters A conceptual model of ice cream consumption (from the IceCream dataset) is... consumption temperature which is represented statistically as... consumption = β 0 + β 1 temperature where

29 GLM the parameters A conceptual model of ice cream consumption (from the IceCream dataset) is... consumption temperature which is represented statistically as... consumption = β 0 + β 1 temperature where β 0 estimates consumption when temperature is zero.

30 GLM the parameters A conceptual model of ice cream consumption (from the IceCream dataset) is... consumption temperature which is represented statistically as... consumption = β 0 + β 1 temperature where β 0 estimates consumption when temperature is zero. β 1 estimates the change in consumption for a unit increase in temperature.

31 The most interesting statistic for us is the parameter that estimates the relationship between temperature and consumption ; the parameter β 1. The formal description of this parameter is...

32 The most interesting statistic for us is the parameter that estimates the relationship between temperature and consumption ; the parameter β 1. The formal description of this parameter is... For a unit increase in X, the estimated change in Y is β 1.

33 The most interesting statistic for us is the parameter that estimates the relationship between temperature and consumption ; the parameter β 1. The formal description of this parameter is... For a unit increase in X, the estimated change in Y is β 1. For a unit increase in temperature, the estimated change in consumption is β 1.

34 The most interesting statistic for us is the parameter that estimates the relationship between temperature and consumption ; the parameter β 1. The formal description of this parameter is... For a unit increase in X, the estimated change in Y is β 1. For a unit increase in temperature, the estimated change in consumption is β 1. The values for the parameters are obtained from the statistical output for the model... β 0 = and β 1 =

35 The following output was obtained by putting the model consumption temperature into the Rcmdr GLM menu (as consumption is continuous, select the identity link from the Gaussian family)... glm(formula = Consumption ~ Temperature, family = gaussian(identity), data = IceCream) Coefficients: Estimate Std. Error (Intercept) Temperature

36 The following output was obtained by putting the model consumption temperature into the Rcmdr GLM menu (as consumption is continuous, select the identity link from the Gaussian family)... glm(formula = Consumption ~ Temperature, family = gaussian(identity), data = IceCream) Coefficients: Estimate Std. Error (Intercept) Temperature These parameters are also represented in the effect display... (use the Rcmdr menu options Models, graphs, effect plot...).

37 Temperature effect plot Consumption Temperature

38 Consumption Temperature effect plot computing β 1 As temperature increases by 40 (30 to 70), consumption increases by (0.3 to 0.425). For a unit increase in temperature, consumption increases by ( ) Temperature

39 Consumption Temperature effect plot Temperature computing β 1 As temperature increases by 40 (30 to 70), consumption increases by (0.3 to 0.425). For a unit increase in temperature, consumption increases by ( ). computing β 0 The value of consumption when temperature = 0. A simple estimation from the graph shows β 0 = 0.2.

40 Categorical explanatory variables... It is useful at this stage to look at categorical explanatory variables. A detailed description of analysing categorical explanatory variables is available in... Hutcheson, G. D. (2011). Categorical Explanatory Variables. Journal of Modelling in Management. 6, 2: research-training.net/reading/jmm6(2)contrastcodingupdated.pdf

41 Categorical explanatory variables... It is useful at this stage to look at categorical explanatory variables. A detailed description of analysing categorical explanatory variables is available in... Hutcheson, G. D. (2011). Categorical Explanatory Variables. Journal of Modelling in Management. 6, 2: research-training.net/reading/jmm6(2)contrastcodingupdated.pdf At a basic level, categorical variables are divided into a number of binary comparisons. Each category is then compared to a specific category (the reference).

42 Categorical explanatory variables... It is useful at this stage to look at categorical explanatory variables. A detailed description of analysing categorical explanatory variables is available in... Hutcheson, G. D. (2011). Categorical Explanatory Variables. Journal of Modelling in Management. 6, 2: research-training.net/reading/jmm6(2)contrastcodingupdated.pdf At a basic level, categorical variables are divided into a number of binary comparisons. Each category is then compared to a specific category (the reference). The following slide shows a model of examination score and whether this is related to the school a child is at. school is an unordered categorical variable with three categories (schoola, schoolb and schoolc).

43 The conceptual model examination score school

44 The conceptual model examination score school is represented statistically as score = β 0 + β 1 schoolb + β 2 schoolc where

45 The conceptual model examination score school is represented statistically as score = β 0 + β 1 schoolb + β 2 schoolc where β 0 estimates score when schoolb and schoolc are both zero (this is equivalent to the score for schoola).

46 The conceptual model examination score school is represented statistically as score = β 0 + β 1 schoolb + β 2 schoolc where β 0 estimates score when schoolb and schoolc are both zero (this is equivalent to the score for schoola). β 1 estimates the change in score for schoolb compared to schoola.

47 The conceptual model examination score school is represented statistically as score = β 0 + β 1 schoolb + β 2 schoolc where β 0 estimates score when schoolb and schoolc are both zero (this is equivalent to the score for schoola). β 1 estimates the change in score for schoolb compared to schoola. β 2 estimates the change in score for schoolc compared to schoola.

48 The following output from the schools.csv dataset was obtained by putting the model score school into the Rcmdr GLM menu (as consumption is continuous, identify the identity link)... glm(formula = SCORE ~ SCHOOL, family = gaussian(identity), data = schools) Coefficients: Estimate Std. Error (Intercept) SCHOOL[T.schoolB] SCHOOL[T.schoolC]

49 The following output from the schools.csv dataset was obtained by putting the model score school into the Rcmdr GLM menu (as consumption is continuous, identify the identity link)... glm(formula = SCORE ~ SCHOOL, family = gaussian(identity), data = schools) Coefficients: Estimate Std. Error (Intercept) SCHOOL[T.schoolB] SCHOOL[T.schoolC] The predicted score at schoola is SchoolB is 7.4 points higher (70.6) and schoolc is 7.6 points lower (56.6).

50 This result can be easily seen in the effect display that accompanies the model SCORE schoola schoolb schoolc SCHOOL

51 This result can be easily seen in the effect display that accompanies the model β 0 = 63.2 SCORE The value of score for schoola (the reference category) schoola schoolb schoolc SCHOOL

52 This result can be easily seen in the effect display that accompanies the model β 0 = 63.2 SCORE The value of score for schoola (the reference category). β 1 = schoolb is 7.4 units higher than schoola. 50 schoola schoolb schoolc SCHOOL

53 This result can be easily seen in the effect display that accompanies the model β 0 = 63.2 SCORE The value of score for schoola (the reference category). β 1 = schoolb is 7.4 units higher than schoola. 50 schoola schoolb schoolc SCHOOL β 2 = 56.6 schoolc is 7.6 units lower than schoola.

54 interpreting parameters for other GLMs The interpretation of the regression coefficients is essentially the same for all GLMs. For example, in order to model the number of checks a suspect appears on and whether this is related to a person s age (these data are from the Arrests dataset from the effects package)... number of checks age

55 interpreting parameters for other GLMs The interpretation of the regression coefficients is essentially the same for all GLMs. For example, in order to model the number of checks a suspect appears on and whether this is related to a person s age (these data are from the Arrests dataset from the effects package)... number of checks age as the number of checks is a count variable, a log-link should be used. The statistical model for this is... log (checks) = β 0 + β 1 age

56 interpreting parameters for other GLMs The interpretation of the regression coefficients is essentially the same for all GLMs. For example, in order to model the number of checks a suspect appears on and whether this is related to a person s age (these data are from the Arrests dataset from the effects package)... number of checks age as the number of checks is a count variable, a log-link should be used. The statistical model for this is... log (checks) = β 0 + β 1 age The parameter β 0 indicates the value of log(checks) when age equals zero. The parameter β 1 indicates the change in log(checks) for a unit change in age.

57 The following output was obtained by putting the model checks age into the Rcmdr GLM menu (as checks is count, identify the log link from the Poisson family)... glm(formula = checks ~ age, family = poisson(log), data = Arrests) Coefficients: Estimate Std. Error (Intercept) age

58 The following output was obtained by putting the model checks age into the Rcmdr GLM menu (as checks is count, identify the log link from the Poisson family)... glm(formula = checks ~ age, family = poisson(log), data = Arrests) Coefficients: Estimate Std. Error (Intercept) age When age = 0, log(count) = For a unit increase in age, log(count) changes by

59 This result can be easily seen in the effect display that accompanies the model... The follow graph shows the relationship between age and log(count) checks age

60 This result can be easily seen in the effect display that accompanies the model... The follow graph shows the relationship between age and log(count) β 0 = (see above) A simple estimation from the graph shows that when age =0, log(checks)=0.15. checks age

61 This result can be easily seen in the effect display that accompanies the model... The follow graph shows the relationship between age and log(count)... β 0 = (see above) A simple estimation from the graph shows that when age =0, log(checks)=0.15. checks age β 1 = (see above) As age increases from 20 to 60 (increase of 40), log(checks) increases from 0.43 to 1 (increase of 0.57). For each unit increase in age, log(checks) increases by 0.57 =

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

63 The model parameters for GLM models have an easy interpretation; one that is essentially the same for all GLM models. The parameters can be interpreted from the standard statistical output shown in the tables, or from the effect displays.

64 Assessing significance...

65 Assessing significance for GLMs In order to interpret the models, it is useful to know the significance associated with each of the parameter estimates. Although consumption may increase by for each unit increase in temperature ; there is no indication whether this increase may have been due to chance.

66 Assessing significance for GLMs In order to interpret the models, it is useful to know the significance associated with each of the parameter estimates. Although consumption may increase by for each unit increase in temperature ; there is no indication whether this increase may have been due to chance. In addition to the parameter estimates, we usually also want to know which parameters, or groups of parameters are significant.

67 Assessing significance for GLMs In order to interpret the models, it is useful to know the significance associated with each of the parameter estimates. Although consumption may increase by for each unit increase in temperature ; there is no indication whether this increase may have been due to chance. In addition to the parameter estimates, we usually also want to know which parameters, or groups of parameters are significant. This is where the GLM models excel, as they employ a simple method for determining significance; one that applies generally to all GLMs.

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

69 Deviance Significance is assessed in GLMs using a common method based on a statistic known as the deviance. The deviance is simply a measure of the difference between the values predicted by the model and the actual values (predicted values compared to the observed data). If the model provides a good prediction of the response variable, the deviance will be relatively small. If the model does not provide accurate predictions of the response variable, the deviance will be relatively large.

70 Deviance Significance is assessed in GLMs using a common method based on a statistic known as the deviance. The deviance is simply a measure of the difference between the values predicted by the model and the actual values (predicted values compared to the observed data). If the model provides a good prediction of the response variable, the deviance will be relatively small. If the model does not provide accurate predictions of the response variable, the deviance will be relatively large. The deviance basically gives an indication of how well the model fits the data.

71 Using the deviance, it is easy to determine the significance of individual and/or groups of variables by comparing nested models. For example, if we wanted to assess whether temperature is a significant predictor of ice cream consumption, we can compare a model of consumption that includes temperature, with a model that does not. consumption = β 0 + β 1 temperature deviance = consumption = β 0 deviance = 0.126

72 Using the deviance, it is easy to determine the significance of individual and/or groups of variables by comparing nested models. For example, if we wanted to assess whether temperature is a significant predictor of ice cream consumption, we can compare a model of consumption that includes temperature, with a model that does not. consumption = β 0 + β 1 temperature deviance = consumption = β 0 deviance = The effect that temperature has had on the model is to reduce the deviance by ( ). All of this information is given in the analysis output...

73 Using the deviance, it is easy to determine the significance of individual and/or groups of variables by comparing nested models. For example, if we wanted to assess whether temperature is a significant predictor of ice cream consumption, we can compare a model of consumption that includes temperature, with a model that does not. consumption = β 0 + β 1 temperature deviance = consumption = β 0 deviance = The effect that temperature has had on the model is to reduce the deviance by ( ). All of this information is given in the analysis output... The Null deviance is the deviance in the response variable without taking into account any other information. The Residual deviance is the deviance in the model that includes the explanatory variables.

74 glm(formula = Consumption ~ Temperature, family = gaussian(identity), data = IceCream) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-09 Temperature e-07 Null deviance: on 29 degrees of freedom Residual deviance: on 28 degrees of freedom Analysis of Deviance Table (Type II tests) Response: Consumption SS Df F Pr(>F) Temperature e-07 Residuals

75 The model above was obtained by running a GLM on the IceCream data and then requesting an analysis of deviance table. These commands can be issued via the Rcmdr menus, or direct to a script file using the commands... GLM.1 <- glm(consumption ~ Temperature, family=gaussian(identity), data=icecream) summary(glm.1) Anova(GLM.1, type="ii", test="f")

76 The model above was obtained by running a GLM on the IceCream data and then requesting an analysis of deviance table. These commands can be issued via the Rcmdr menus, or direct to a script 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. A detailed description of significance tests and deviance is provided in... Hutcheson, G. D. and Moutinho, L. (2008). Statistical Modelling for Management. Sage Publications.

77 Temperature effect plot Consumption Temperature

78 Consumption Temperature effect plot Temperature significance of temperature a visual indication of the significance of temperature can be seen in the effect display. It is easy to see from the line and associated confidence intervals shown in the graph (the shaded area of the plot) that predictions of consumption are different as temperature changes. Information about temperature is, therefore, important for predicting consumption.

79 The significance of temperature in the model can be manually calculated by comparing the nested models... model01: consumption = β 0 model02: consumption = β 0 + β 1 temperature This can be easily achieved using the Rcmdr Models, Hypothesis tests, Compare two models... menu option.

80

81 The output clearly shows which models are being compared. Note that the statistics for Temperature are the same as those provided previously... 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)

82 The same underlying theory may be applied to testing categorical explanatory variables and different response variables. The following example shows a binary response variable being predicted by a categorical and a continuous explanatory variable. These data are from the Arrests dataset in the effects package (to reproduce these results, don t forget to change the variable year to a categorical variable yearcat ).

83 The same underlying theory may be applied to testing categorical explanatory variables and different response variables. The following example shows a binary response variable being predicted by a categorical and a continuous explanatory variable. These data are from the Arrests dataset in the effects package (to reproduce 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

84 The same underlying theory may be applied to testing categorical explanatory variables and different response variables. The following example shows a binary response variable being predicted by a categorical and a continuous explanatory variable. These data are from the Arrests dataset in the effects package (to reproduce 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 which is represented statistically as... logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year β 6 age

85 or represented more succinctly as... logit(released) = β y=5 β y(1 5) yearcat + β 6 age

86 or represented more succinctly as... logit(released) = β y=5 β y(1 5) yearcat + β 6 age The categorical variable yearcat is represented using 5 binary comparisons; each year is compared to the reference category.

87 or represented more succinctly as... logit(released) = β y=5 β y(1 5) yearcat + β 6 age The categorical variable yearcat is represented using 5 binary comparisons; each year is compared to the reference category. A GLM (using the logit link) is shown below...

88 glm(formula = released ~ age + yearcat, family = binomial(logit), data = Arrests) Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) < 2e-16 age yearcat[t.1998] yearcat[t.1999] yearcat[t.2000] e-06 yearcat[t.2001] yearcat[t.2002] Null deviance: on 5225 degrees of freedom Residual deviance: on 5219 degrees of freedom

89 The logit model of released shows the deviance scores at the bottom of the tabular output. The Null deviance tells us how much deviance there is in the variable released (an empty model) and the Residual deviance tells us how much deviance there is in the model with all the explanatory variables included.

90 The logit model of released shows the deviance scores at the bottom of the tabular output. The Null deviance tells us how much deviance there is in the variable released (an empty model) and the Residual deviance tells us how much deviance there is in the model with all the explanatory variables included. In order to find out if each variable is significant, we need to look at the analysis of deviance table... Analysis of Deviance Table (Type II tests) Response: released LR Chisq Df Pr(>Chisq) age ** yearcat ***

91 From the Analysis of Deviance table, we can see that including yearcat in the model results in a reduction in deviance of This is the result of comparing the deviance of the following models... and logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year β 6 age logit(released) = β 0 + β 6 age

92 From the Analysis of Deviance table, we can see that including yearcat in the model results in a reduction in deviance of This is the result of comparing the deviance of the following models... and logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year β 6 age logit(released) = β 0 + β 6 age This change in deviance is significant and provides evidence that the variable yearcat may be influential in predicting whether or not someone is released.

93 These models can be compared using the Rcmdr Models, Hypothesis tests, Compare two models... menu option. Analysis of Deviance Table Model 1: released ~ age Model 2: released ~ yearcat + age Resid. Df Resid. Dev Df Deviance Pr(>Chi)

94 Including the variable age in the model results in a reduction in deviance of This is the result of removing 1 parameter from the model (df = 1). and logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year β 6 age logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year2002

95 Including the variable age in the model results in a reduction in deviance of This is the result of removing 1 parameter from the model (df = 1). and logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year β 6 age logit(released) = β 0 + β 1 year β 2 year β 3 year β 4 year β 5 year2002 This change in deviance is significant and provides evidence that the variable age may be influential in predicting whether or not someone is released.

96 These models can be compared using the Rcmdr Models, Hypothesis tests, Compare two models... menu option. Analysis of Deviance Table Model 1: released ~ yearcat Model 2: released ~ yearcat + age Resid. Df Resid. Dev Df Deviance Pr(>Chi)

97 These results can also be inferred from the effect displays. Although these do not provide estimates of significance, the graphics provide enough information to identify the important relationships visually. yearcat effect plot age effect plot 0.88 released released yearcat age

98 The significance of each category to the prediction of the response variable in logit models is estimated using the z-distribution in the standard regression output. It should be noted that this is a large-sample approximation and the deviance statistic is preferable for assessing significance. Full information about this is available in... Hutcheson, G. D. and Moutinho, L. (2008). Statistical Modelling for Management. Sage Publications.

99 Type II and Type III ANOVA tests The choice of ANOVA test that is used to compare models is very important, particularly for models with interactions.

100 Type II and Type III ANOVA tests The choice of ANOVA test that is used to compare models is very important, particularly for models with interactions. Type III ANOVA tests are those that are computed for individual parameters and are provided in the standard tabular output.

101 Type II and Type III ANOVA tests The choice of ANOVA test that is used to compare models is very important, particularly for models with interactions. Type III ANOVA tests are those that are computed for individual parameters and are provided in the standard tabular output. Type II ANOVA tests are those that are computed for variables and are provided in the Analysis of Deviance table. The difference between the two types of test is very important and will be highlighted using examples later in the course. The difference between the two can be illustrated using the following model (from the ICEcream data) which contains an interaction... consumption temperature income

102 consumption temperature income Type III tests Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 1.216e e Temperature e e Income 7.459e e Temperature:Income 6.250e e Type II tests SS Df F Pr(>F) Temperature e-08 *** Income ** Temperature:Income

103 The significance of the variables temperature and Income are different depending on the type of test used. The type III test for temperature compares the models... consumption = β 0 + β 1 temperature + β 2 income + β 3 temperature:income and consumption = β 0 + β 2 income + β 3 temperature:income

104 The significance of the variables temperature and Income are different depending on the type of test used. The type III test for temperature compares the models... consumption = β 0 + β 1 temperature + β 2 income + β 3 temperature:income and consumption = β 0 + β 2 income + β 3 temperature:income whilst the type II test for temperature compares the models... consumption = β 0 + β 1 temperature + β 2 income and consumption = β 0 + β 2 income

105 The type III test does not provide an indication of the significance of the variable temperature. We need to compare a model that includes temperature with a model thatdoes not - BOTH of these modesl include temperature.

106 The type III test does not provide an indication of the significance of the variable temperature. We need to compare a model that includes temperature with a model thatdoes not - BOTH of these modesl include temperature. The type II test does compare models differentiated on the basis of temperature and are, therefore, more appropriate for assessing the effect of temperature.

107 Traditional tests and equivalent GLM models...

108 Traditional tests and equivalent GLM models The GLM models reproduce or replace many of the traditional tests. For example, tests for independent group designs...

109 Traditional tests and equivalent GLM models The GLM models reproduce or replace many of the traditional tests. For example, tests for independent group designs... Traditional Test GLM one independent variable t-test (unrelated) Mann-Whitney 1-way ANOVA (unrelated) Kruskal-Wallis Jonck-heere Trend chi-square (contingency table) etc., etc. Y X multiple independent variables complex selection of multi-way ANOVA models multi-way contingency tables (log-linear) Y X 1 + X 2

110 Traditional tests and equivalent GLM models... and tests for dependent (or matched) group designs... Traditional Test GLM one independent variable paired t-test Wilcoxon 1-way ANOVA (related) Friedman Pages L-trend etc., etc., Y subject + X multiple independent variables complex selection of multi-way ANOVA models multi-way contingency tables (log-linear) Y subject + X 1 + X 2

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

112 In order to realise the power of the GLMs, it is important to understand 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: http: //

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

114 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation

115 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation 2. Identify scale of measurement for the response variable

116 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation 2. Identify scale of measurement for the response variable 3. Select the appropriate model in the Rcmdr.

117 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation 2. Identify scale of measurement for the response variable 3. Select the appropriate model in the Rcmdr. Generalized linear model... for continuous and count responses

118 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation 2. Identify scale of measurement for the response variable 3. Select the appropriate model in the Rcmdr. Generalized linear model... for continuous and count responses Multinomial logit model... for unordered categorical responses

119 The usefulness of the equation format for representing statistical models is evident when the analyses are run in the software. In order to select the appropriate GLM define equation 2. Identify scale of measurement for the response variable 3. Select the appropriate model in the Rcmdr. Generalized linear model... for continuous and count responses Multinomial logit model... for unordered categorical responses Ordinal regression model... for ordered categorical responses

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121 Then enter the equation...

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126 Exercises...

127 Your research model is... salary age + qualification where salary and age are continuous and qualification is recorded in 4 ordered categories. 1. which link function is appropriate for this model? 2. How many parameters would you expect to see in the model for qualification? 3. which statistical technique would you use? 4. what is the statistical model (the linear model)?

128 Your research model is... A-level IQ + gender where A-level is recorded as 6 ordered categories, IQ is recorded as continuous, and gender is recorded as 2 unordered categories. 1. which link function is appropriate for this model? 2. How many parameters would you expect to see in the model for IQ? 3. which statistical technique would you use? 4. what is the statistical model (the linear model)?

129 Your research model is... traffic violations age gender where traffic violations is recorded as the number of recorded violations (0 to 7), age is recorded as continuous, and gender is recorded as 2 unordered categories. 1. which link function is appropriate for this model? 2. How many parameters would you expect to see in the model (including the intercept)? 3. which statistical technique would you use? 4. what is the statistical model (the linear model)? 5. Would you use a TYPE II or a TYPE III test to ascertain if age is significant?

130 Your research model is... Holiday destination age gender where Holiday destination is recorded as 6 unordered categories, age is recorded as continuous, and gender is recorded as 2 unordered categories. 1. which link function is appropriate for this model? 2. which statistical technique would you use? 3. Would you use a TYPE II or a TYPE III test to ascertain if gender is significant?