Parametric Modelling of Over-dispersed Count Data. Part III / MMath (Applied Statistics) 1
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1 Parametric Modelling of Over-dispersed Count Data Part III / MMath (Applied Statistics) 1
2 Introduction Poisson regression is the de facto approach for handling count data What happens then when Poisson models do not fit the data well? In particular, when there is extra-poisson variation? That is, over-dispersion? Which may be due to Omission of important covariates Misspecification of covariate-response relationships Non-independence between events making up a count (contagion) Natural heterogeneity (unmeasured/unmeasurable covariates) Excess zeroes Part III / MMath (Applied Statistics) 2
3 . Over-dispersion By over-dispersion, we mean that the observed variance is larger than the theoretical variance given by the model fitted For count data, under the Poisson assumption, we would expect the empirical variance to be equal to its mean However, there are many situations in practice, where this theoretical mean-variance relationship is not satisfied var(y) > E(Y) Reflected in the Pearson s Chi-squared statistic (or deviance) being much larger than the residual degrees of freedom Failure to account for this extra-poisson variation may lead to estimated standard errors being incorrectly too precise Misleading inferences Part III / MMath (Applied Statistics) 3
4 Accounting for Over-dispersion Approaches available to account for over-dispersion Additive and Generalised Additive Models used to model non-linear relationships (?) Quasi-likelihood used when it is important to model correctly the mean-variance relationship. Not likelihood-based! Continuous mixture models when unobserved heterogeneity (and/or contagion) is thought to explain the over-dispersion and correctly modelling the mean-variance relationship is important Discrete mixture models used to account for (apparent) excess zeroes not able to be captured by the Poisson assumption Part III / MMath (Applied Statistics) 4
5 Negative Binomial Models Negative binomial models are natural extensions of Poisson models when there is unobserved heterogeneity which can be represented continuously Instead of assuming that the mean parameter, μ, in the Poisson model is a deterministic function of the covariates, we let μ be random Specifically, we let μ = ηυ, with log η = β T x and υ~gamma( θ, θ ) Conditional on μ, Y μ ~ Poisson( μ) Part III / MMath (Applied Statistics) 5
6 Marginal distribution of Y Negative binomial f( Y ηθ, ) y θ Γ ( θ + y) η θ = Γ θ ( θ) y! ( η + θ ) + y The mean and variance are E( Y) = η and var( Y) = η+ η = η+ τη θ τ is the over-dispersion parameter τ = 0 indicates no heterogeneity Poisson model Part III / MMath (Applied Statistics) 6
7 Negative Binomial Regression Estimation of regression parameters, β, and overdispersion parameter, τ (or alternatively θ), proceeds as per usual (i.e. maximum likelihood estimation) Assuming τ > 0 and under certain regularity conditions, mle s of the regression parameters and over-dispersion parameter will be asymptotically normal Additionally, the mle of β will be asymptotically independent of the mle of τ Part III / MMath (Applied Statistics) 7
8 Negative Binomial Regression Hypothesis testing of τ = 0 (i.e. homogeneity) is nonstandard as falls on the boundary of the parameter space LRT (NB to Pois) has an asymptotic 50:50 mixture dist n of point mass at 0 and χ 2 -dist n with 1 df Under null hypothesis that τ = 0, the Wald test statistic, Z, has an asymptotic 50:50 mixture dist n of point mass at 0 and a half-normal distribution For τ > 0, standard large sample theory apply and hypothesis testing of the regression parameters, β, proceeds as usual Part III / MMath (Applied Statistics) 8
9 Zero-Inflated Models When over-dispersion is thought to be due to excessive number of zeroes (i.e. more zeroes than expected) Some of the zeroes observed may be sampling zeroes, but others may be structural Example If we ask adults who have just returned to their fishing village from the sea: How many wild salmon did you catch? For those who answer 0, this could be because Unlucky (sampling zero) Did not go fishing! (structural zero) Part III / MMath (Applied Statistics) 9
10 Zero-Inflated Models Thus believe that some of the zeroes (i.e. structural zeroes) belong to a different component than the rest of the data (i.e. sampling zeroes and positive counts) However we cannot (in general) distinguish the sampling zeroes from the structural zeroes Results in a two component mixture model: π + (1 π) g(0 μ) for y = 0 Pr( Y = y) = (1 π) g( y μ) for y > 0 Part III / MMath (Applied Statistics) 10
11 Zero-Inflated Models π is the probability of being a structural zero (i.e. belonging to the first component) g(y μ) is the probability mass function for belonging to the second component and typically chosen to be either from a Poisson or a negative binomial Either get a zero-inflated Poisson (ZIP) model or a zeroinflated negative binomial (ZINB) model Mean and variance of ZIP are E( Y μ, π) = (1 π) μ and var( Y μ, π) = (1 π) μ(1 + πμ) Mean and variance of ZINB are E( Y μ, π) = (1 π) μ and var( Y μ, π) = (1 π) μ(1 + ( π + τ) μ) Part III / MMath (Applied Statistics) 11
12 Zero-Inflated Regression Covariates are allowed to enter both components of the model π log = γ 1 π log μ = β T x But need not have the same set of covariates in both components Maximum likelihood estimation may proceed by directly maximising the log-likelihood or through use of the EM algorithm T z Part III / MMath (Applied Statistics) 12
13 Model Comparison Assessment of regression parameter significance is straightforward However, we may additionally be interested in comparing either ZIP v Poisson or ZINB v negative binomial But the models to be compared are not nested Standard LRT s are not appropriate Can use either AIC or BIC instead Alternatively can use the Vuong Test (Vuong, 1989) Details of Vuong Test given in lecture notes! Greene (1994) recommends the use of the Vuong Test Part III / MMath (Applied Statistics) 13
14 Two-Part (Hurdle) Models Alternative to zero-inflated models for handling excess zeroes Again based on two separate data generating processes or mechanisms The first data generating process (or first part) determines whether or not the response is zero The second data generating process (or second part) determines the actual count if non-zero In this two-part framework we specify the probability, π, of generating a non-zero count (i.e. the probability of clearing the hurdle ) the distribution for the non-zero counts, after clearing the hurdle Part III / MMath (Applied Statistics) 14
15 Mathematically, Two-Part (Hurdle) Models (1 π ) for y = 0 Pr( Y = y) = g( y μ) π for y > 0 (1 g(0 μ)) Typically, g(..) chosen to be the pmf of either a Poisson or negative binomial with mean μ Again in regression setting, we can introduce covariates into both parts of the two-part model: π T T log = γ z and log μ = β x 1 π Part III / MMath (Applied Statistics) 15
16 Two-Part (Hurdle) Regression If the sets of regression parameters are disjoint then we can estimate the parameters in each part by considering separately the two parts of the model That is, separate analyses are performed for the proportion of non-zero counts (e.g. logistic regression); and for the positive counts (e.g. truncated Poisson regression) This is, of course, because the full likelihood factorizes If parameters are shared across the two sets then direct maximisation of the log-likelihood would be required Comparison of nested or non-nested models can proceed as mentioned earlier Part III / MMath (Applied Statistics) 16
17 Conclusion Discussed over-dispersion (relative to Poisson) How to accommodate over-dispersion when observed Through use of continuous or discrete mixture models For these models, we need to be aware of the interpretation of the regression parameters In introducing these models, we briefly discussed estimation, inference and model choice Important to be aware of the fact that the approach adopted to account for over-dispersion in count data may depend on subject-matter knowledge and not just on statistical considerations Part III / MMath (Applied Statistics) 17
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