Wronged Questions: Non-linear Models

Question 1

T/F: Pearson’s goodness-of-fit test statistic is normally distributed under the null hypothesis.

Answer 1

False. It follows the chi-squared distribution

Question 2

T/F: The number of degrees of freedom for the likelihood ratio test is the number of parameters in the reduced model.

Answer 2

False. The number of DF is determined by the # of variables removed from the full model to create the reduced model

Question 3

T/F: Akaike information criterion values are better when they are smaller, as they indicate a more parsimonious model.

Answer 3

True. We want a small AIC value

Question 4

T/F: Z-statistics are not used in Poisson regression for testing the significance of individual regression coefficients.

Answer 4

False. They are used for this purpose

Question 5

T/F: The Bayesian information criterion generally favors models with more parameters over simpler ones.

Answer 5

False. BIC favours simpler models

Question 6

T/F: A square root link is as appropriate as a log link.

Answer 6

False. A square root link outputs a non-negative number. It’s better if a link function outputs a real number.

Question 7

T/F: If the GLM is adequate, the deviance is a realization from a chi-square distribution.

Answer 7

True. If a GLM is adequate, the scaled deviance is a realization from a chi-square distribution because it equals twice the difference of maximized log-likelihoods for nested models. For a Poisson regression, the deviance also equals the scaled deviance.

Question 8

Hurdle Model

Answer 8

Model where a random variable is modelled using two parts.
1) Probability of obtaining 0
2) Probability of obtaining a non-zero probability

Question 9

Log link function ensures predicted values are always _____, suitable for count data.

Answer 9

Positive

Question 10

Log link isn’t standard for ______ distribution

Answer 10

Binomial

Question 11

Gamma distribution is for positive, _________ data

Answer 11

Continuous

Question 12

Identity link doesn’t guarantee ________ predictions for Poisson.

Answer 12

Positive

Question 13

Logit link maps the linear predictor N to the interval _____, which is appropriate for probabilities

Answer 13

(0, 1)

Question 14

The link function in a GLM is a function that relates the linear predictor to the ____ of the distribution.

Answer 14

mean

Question 15

T/F: When using a linear probability model with a binary response variable, the main advantage is the relative ease of parameter interpretation.

Answer 15

True. A linear probability model, despite its limitations, is favored for its straightforward interpretability. The coefficients in a linear probability model can be directly interpreted as changes in probability associated with unit changes in predictors.

Question 16

T/F: It is easy to distinguish between logit and probit models graphically, since the forms of their functions are quite different.

Answer 16

False. Graphically, they are not easily distinguishable because both functions are sigmoid (S shaped) and very close in form.

Question 17

T/F: Between logit and probit models, one is significantly more popular because its cumulative distribution function is the only one of the two that has a closed-form expression.

Answer 17

True. The logit model tends to be more popular in many applications primarily because the logistic distribution used in the logit model has a closed-form cumulative distribution function.

Question 18

T/F: Both logit and probit models have functions pi(z) that must be between 0 and 1, as they are used to model probabilities.

Answer 18

True. Both models are designed to model probabilities, and thus, the output of their respective functions, pi(z), must be bounded between 0 and 1.

Question 19

T/F: Both logit and probit models aim to circumvent the disadvantages of linear probability models.

Answer 19

True. Both models are used to overcome limitations seen in linear probability models, particularly issues related to heteroscedasticity and probabilities being modeled outside the [0, 1] interval.

Question 20

T/F: The dependent variable in Poisson regression is a non-negative integer that counts the number of events.

Answer 20

True. This statement is correct as Poisson regression is used specifically for count data, which are non-negative integers.

Question 21

T/F: The maximum likelihood estimator for the mean in Poisson regression with no predictors is the sample mean of the observed counts.

Answer 21

True. The maximum likelihood estimator (MLE) for the mean in a Poisson distribution is indeed the sample mean of the counts.

Question 22

T/F: The logarithmic link function is used to connect the mean of the dependent variable to the explanatory variables.

Answer 22

True. The logarithmic link function is a standard component in Poisson regression to relate the mean to the explanatory variables.

Question 23

T/F: Poisson regression can incorporate exposures as an explanatory variable to allow the mean to vary with known amounts.

Answer 23

True. It is correct that exposures can be incorporated into the Poisson regression model to allow variations in the mean.

Question 24

T/F: OLS assumes that the response variable is continuous and normally distributed, while GLMs can accommodate various types of distributions like binomial, Poisson, and normal.

Answer 24

True. OLS typically assumes the response variable is continuous and normally distributed. In contrast, GLMs are designed to handle various types of distributions through their distribution family.

Question 25

T/F: GLMs include a link function that relates the mean of the response variable to the linear predictor, whereas OLS directly models the mean of the response variable as a linear combination of predictors.

Answer 25

True. GLMs use a link function to connect the mean of the response variable to the linear predictors, which is crucial for dealing with non-normal distributions. OLS, however, does not use a link function and models the response variable directly as a linear combination of predictors without transformation.

Question 26

T/F: Both OLS and GLM assume that observations are independent of each other.

Answer 26

True. The assumption of independent observations is fundamental in both OLS and GLMs. The independence of observations is crucial for the validity of the statistical tests used in inference, such as tests for coefficients and predictions.

Question 27

T/F: GLMs do not require homoscedasticity, unlike OLS which assumes homoscedasticity.

Answer 27

True. GLMs can handle cases where the variance of the response variable is not constant (heteroscedasticity), due to the nature of the probability distributions used. OLS, on the other hand, assumes constant variance across all levels of the predictor variables (homoscedasticity).

Question 28

T/F: OLS assumes that the residuals are normally distributed, while GLM assumes that the residuals are uniformly distributed.

Answer 28

False. OLS assumes that the residuals are normally distributed. GLM does not focus on residuals and only focuses on the distribution (gamma, gaussian, etc) of the variable itself.

Question 29

T/F: The three major drawbacks of the linear probability model are poor fitted values, heteroscedasticity, and meaningless residual analysis.

Answer 29

True

Question 30

T/F: The logistic and probit regression models aim to circumvent the drawbacks of linear probability models.

Answer 30

True

Question 31

T/F: The logit and probit functions are substantially different.

Answer 31

False.

Question 32

T/F: A large Pearson chi-square statistic indicates that overdispersion is likely more severe.

Answer 32

True. A large Pearson chi-square statistic suggests that delta should have a large estimate, thus indicating a big change is necessary to address overdispersion.

Question 33

Anscombe residuals

Answer 33

Used for non-normal response distributions in GLM because Pearson is often skewed.

Question 34

Pearson Residual

Answer 34

Question 35

Deviance statistic

Answer 35

Question 36

Scaled deviance

Answer 36

Question 37

X^2 statistic for likelihood ratio test

Answer 37

Dr* - Df*

Question 38

Pseudo r^2

Answer 38

Question 39

Canonical link b'^-1(μ) for inverse gaussian distribution

Answer 39

Question 40

Canonical link b'^-1(μ) for gamma distribution

Answer 40

Question 41

Canonical link b'^-1(μ) for negative binomial distribution

Answer 41

Question 42

Canonical link b'^-1(μ) for poisson distribution

Answer 42

ln(μ)

Question 43

Canonical link b'^-1(μ) for binomial distribution

Answer 43

Question 44

Canonical link b'^-1(μ) for gaussian distribution

Answer 44

μ

Question 45

b(θ) for Gaussian distribution

Answer 45

Question 46

b(θ) for binomial distribution

Answer 46

Question 47

b(θ) for poisson distribution

Answer 47

Question 48

b(θ) for negative binomial distribution

Answer 48

Question 49

b(θ) for gamma distribution

Answer 49

-ln(-θ)

Question 50

b(θ) for inverse gaussian distribution

Answer 50

Question 51

Φ for gaussian distribution

Answer 51

σ^2

Question 52

Φ for binomial distribution

Answer 52

1

Question 53

Φ for poisson distribution

Answer 53

1

Question 54

Φ for negative binomial distribution

Answer 54

1

Question 55

Φ for gamma distribution

Answer 55

1/α or α^-1

Question 56

Φ for inverse gaussian distribution

Answer 56

1/λ or λ^-1

Question 57

θ for gaussian distribution

Answer 57

μ

Question 58

θ for binomial distribution

Answer 58

Question 59

θ for poisson distribution

Answer 59

ln(λ)

Question 60

θ for negative binomial distribution

Answer 60

ln(1-p)

Question 61

θ for gamma distribution

Answer 61

-γα^-1

Question 62

θ for inverse gaussian distribution

Answer 62

Question 63

PDF of gaussian distribution

Answer 63

Question 64

PDF of binomial distribution

Answer 64

(n x) part is n!/(n-x)!x!

Question 65

PDF of poisson distribution

Answer 65

Question 66

PDF of negative binomial distribution

Answer 66

Question 67

PDF of gamma distribution

Answer 67

Question 68

PDF of inverse gaussian distribution

Answer 68

Question 69

Var(X) of gamma distribution

Answer 69

Φ x μ^2

Question 70

A heterogeneity model requires modeling with a _________ mixture.

Answer 70

Continuous

Question 71

A zero-inflated model requires modeling as a mixture of a point mass at zero and another distribution whose domain starts with _.

Answer 71

0

Question 72

For ΦE[Y]Y^p, what is p for gaussian distribution?

Answer 72

0

Question 73

For ΦE[Y]Y^p, what is p for poisson distribution?

Answer 73

1

Question 74

For ΦE[Y]Y^p, what is p for tweeide/compound poisson-gamma distribution?

Answer 74

(1,2)

Question 75

For ΦE[Y]Y^p, what is p for gamma distribution?

Answer 75

2

Question 76

For ΦE[Y]Y^p, what is p for inverse gaussian distribution?

Answer 76

3

Question 77

T/F: The cumulative probit model uses cumulative probabilities that differ significantly from those used in the cumulative logit model.

Answer 77

False. The cumulative probit model and the cumulative logit model both use similar structures of cumulative probabilities.

Question 78

T/F: The proportional odds model uses logit[Pr(yi<=j)] = aj, which is an intercept-only model and avoids using any explanatory variables.

Answer 78

False. The proportional odds model actually incorporates explanatory variables, and is expressed as logit[Pr(yi<=j)] = aj + Xi^TB. This model is not intercept-only.

Question 79

T/F: In the simplest cumulative logit model, the cut-point parameters are decreasing.

Answer 79

False. In the simplest cumulative logit model, the cut-point parameters are non-decreasing reflecting the ordered nature of the probabilities. This also applies in a proportional odds model.

Question 80

T/F: Ordinal variables are unordered categorical variables that do not consider the sequence of categories.

Answer 80

False. Ordinal variables are ordered categorical variables, where the sequence of categories is significant and often represents a scale of measurement.

Question 81

T/F: The cumulative probit model will give results that are very similar to the cumulative logit model.

Answer 81

True. The cumulative probit model and the cumulative logit model both predict ordinal dependent variables and are expected to give similar results primarily because they share a similar conceptual framework, despite using different link functions.

Question 82

T/F: The link function for incorporating exposures explicitly is ln(μi) = ln(Ei) + Xi^Tβ.

Answer 82

True

Question 83

T/F: If exposures are not considered, the mean of the Poisson distribution is modeled as μi = exp(xi^Tβ).

Answer 83

True

Question 84

T/F: The likelihood ratio test is used for testing groups of regression coefficients.

Answer 84

True. The likelihood ratio test (LRT) is indeed used to compare the fit of two nested models, specifically testing whether the simpler (or reduced) model is sufficient or the full model with additional parameters is justified.

Question 85

T/F: The degrees of freedom for the likelihood ratio test correspond to the number of variables included in the full model.

Answer 85

False. The df for the LRT is determined by the number of variables removed from the full model to form the reduced model.

Question 86

T/F: The simplest cumulative logit model is logit[Pr(yi<=j)] = aj, which is an model that is intercept-only and does not use any of the explanatory variables.

Answer 86

True

Question 87

T/F: For the cumulative logit model, we use logit[Pr(yi<=j)] = aj, where the parameter interpretation is similar to logistic regression.

Answer 87

True

Question 88

T/F: Both the cumulative logit and cumulative probit models are based on cumulative probabilities.

Answer 88

True

Question 89

T/F: The proportional odds model does not incorporate explanatory variables.

Answer 89

False. The proportional odds model does incorporate explanatory variables.

Question 90

T/F: Ordinal variables are ordered categorical variables where the order of categories matters, representing a scale from smallest to largest or vice versa.

Answer 90

True. Ordinal variables are indeed ordered categories where the order is significant, depicting a sequence or scale which reflects increasing or decreasing magnitude or importance.

Question 91

T/F: Pearson residuals can be used to calculate a goodness-of-fit statistic.

Answer 91

True

Question 92

T/F: Pearson residuals can be used to detect if additional variables of interest can be used to improve the model specification.

Answer 92

True

Question 93

T/F: The variance of the response in a GLM is the scale parameter times the variance function v(μ).

Answer 93

True