U.U.D.M. Project Report 2015:9 On Statistical Methods for Zero-Inflated Models Julia Eggers Examensarbete i matematik, 15 hp Handledare och examinator: Silvelyn Zwanzig Juni 2015 Department of Mathematics Uppsala University Abstract Data with excess zeros arise in many contexts. Conventional probability dis- tributions often cannot explain large proportions of zero observations. In this paper we shall study statistical models which take large proportions of zero observations into account. We will consider both discrete and continuous dis- tributions. Contents 1 Introduction 2 2 Models for Zero-Inflated Data 7 3 Models for Semicontinuous Data with Excess Zeros 9 3.1 Tobit Models . 9 3.2 Sample Selection Models . 10 3.3 Double Hurdle Models . 11 3.4 Two-Part Models . 12 4 Inference in Models for Zero-Inflated Data 13 4.1 The Likelihood Function . 14 4.2 Maximum Likelihood Estimators . 14 4.3 Moment Estimators . 15 4.4 Cold Spells in Uppsala . 16 4.5 Exponential Family . 18 5 Inference in Two-Part Models 19 5.1 The Likelihood Function . 20 5.2 Maximum Likelihood Estimators . 20 5.3 Moment Estimators . 21 5.4 Exponential Family . 23 5.5 Hypothesis Testing . 24 References 26 1 1. Introduction In this paper we will study models for data with a large proportion of zeros. For this we will introduce a few terms. Definition 1.0.1. Discrete probability distributions with a large probability mass at zero are said to be zero-inflated. Conventional distributions usually cannot explain the large proportion of zeros in zero-inflated data. For this reason different models which can account for a large proportion of zero observations must be applied instead. Definition 1.0.2. Probability distributions which are continuous on the entire sample space with the exception of one value at which there is a positive point mass are said to be semicontinuous. In this paper we will study models for zero-inflated distributions as well as for semicontinuous distributions with a positive probability mass at zero. We will only consider distributions with non-negative support. Remark 1.0.1. Unlike in the case of left-censored data, zeros in semicontin- uous data correspond to actual observations and do not represent negative or missing values which have been coded as zero. Data with excess zeros may arise in many different contexts. We will start by giving a few examples. Examples of zero-inflated data • Cold spells In his paper on trends for warm and cold spells in Uppsala, Jesper Rydén studied the yearly number of cold spells in Uppsala, Sweden for the period from 1840 to 2012 [07]. He defined a cold spell as a period of at least six consecutive days during which the daily minimum temperature was less 2 than −13:4◦C. The threshold of −13:4◦C was chosen as it corresponds to the 5%-quantile of daily minimum temperatures for the reference period 1961 - 1990. The yearly number of cold spells in Uppsala appears to be zero-inflated as can be seen from the data below. There is a large proportion of zero observations, i.e. years during which there were no cold spells. Figure 1.1: Yearly number of cold spells in Uppsala, 1840 - 2012 • Defects in manufacturing In manufacturing processes defects usually only occur when manufactur- ing equipment is not properly aligned. If the equipment is misaligned, defects can be found to occur according to a Poisson distribution [08]. This implies that defects in manufacturing occur according to a Poisson distribution with inflation at zero. Examples of semicontinuous data with excess zeros • Household expenditures on durable goods The amount of money a household spends monthly on certain durable goods such as cars or appliances like washing machines or refrigerators is distributed according to a semicontinuous distribution. During most months no such goods are purchased and the expense is zero. If durable goods are purchased, the household expenditure on durable goods for that 3 month amounts to some positive value, namely the price of the purchased items. • Alcohol consumption Consider the alcohol consumption of a population during a certain pe- riod of study. Some people belonging to the population may not drink any alcohol at all, thus consuming zero liters of alcohol. These people account for a point mass at zero. People who do consume alcohol may consume arbitrarily large, but positive, amounts. Thus we have a contin- uous distribution for positive values. Similarly, the tobacco consumption or consumption of drugs in general is semicontinuously distributed. • Insurance benefits The Swedish Social Insurance Agency ’Forsäkringskassan’ publishes an- nual reports on its expenditures. The publication ’Social Insurance in Figures 2014’ states that in 2013 a total amount of approximately 24.1 million SEK were paid out as sickness benefits. These sickness benefits are meant to compensate insured for the inability to work due to illness. In 2013, 532 450 people in Sweden received sickness benefits. This corre- sponds to around 9% of all insured between the ages of 16 and 64. The amounts of sickness benefits paid out to insured during the year 2013 are semicontinuously distributed. 91% of all insured received no such ben- efits. We thus have a positive probability mass at zero. Those people who did receive sickness benefits got positive amounts which varied according to factors like income and time spent on sick leave. Therefore, we have a continuous distribution for positive values. Table 1.1 below gives an account of the average amounts of sickness ben- efits paid out to insured depending on gender and age group. 4 Table 1.1: Sickness benefits, 2013 We know from the data that 532 450 insured received sickness benefits in 2013 while around 5 383 661 insured received no such benefits. Since 24.1 million SEK were paid out in total, the average positive amount that was paid out per person that year amounted to approximately 44 822 SEK. Assuming that the paid out benefits are exponentially distributed with parameter λ given that they are positive, we may estimate λ^ = 1=44822. Generating a sample from this distribution in R, we may illustrate how the sickness benefits may have been distributed. Figure 1.2: A possible distribution of the amount of sickness benefits paid out to insured during the year 2013 5 • Healthcare expenditures Healthcare expenditures in general can be found to be semicontinuously distributed. Individuals may or may not choose to seek medical treatment during a certain period of study. There are no costs arising for people who do not seek medical treatment. If medical treatment is sought, however, the cost for the treatment amounts to some positive value. Thus health- care expenditures are continuously distributed for positive values and have a positive probability mass at zero. 6 2. Models for Zero-Inflated Data The models for zero-inflated data which we will present here are variations of the following mixture model Y = ∆Z1 + (1 − ∆)Z2 Z1 Z2 with ∆ ∼ Ber(p), Z1 ∼ P and Z2 ∼ P . Z2 If we let P = δf0g and assume Z1 to be discrete, we obtain a model for zero- inflated data. When modeling count data we have the additional assumption that P (Z1 ≥ 0) = 1. Z1 For the above model we have that Y ∼ δf0g with probability 1−p and Y ∼ P with probability p. Letting pZ1 denote the probability mass function of the random variable Z1 we obtain 1 − p + pp (0) ; y = 0 P (Y = y) = Z1 ppZ1 (y) ; y > 0 When modeling count data, the negative binomial and the Poisson distribu- tion are common distributions for Z1. If Z1 ∼ P o(λ) the above model is referred to as the zero-inflated Poisson model, abbreviated ZIP. Zero-Inflated Poisson Regression Zero-inflated Poisson regression is an extension of the zero-inflated Poisson model which was proposed by Diane Lambert in 1992 [08]. The model assumes Y = (Y1;:::;Yn) to be a sample of independent, but not necessarily identically distributed random variables Yi. In this model we assume Yi ∼ P o(λi) with probability pi. ( 1 − pi + pi exp(−λi) ; yi = 0 Thus P (Y = y ) = yi i i λi exp(−λi) pi ; yi > 0 yi! The parameters p = (p1; : : : ; pn) and λ = (λ1; : : : ; λn) are assumed to satisfy 7 p log(λ) = Bβ and logit(p) = log( 1−p ) = Gγ with B and G denoting matrices with explanatory variables. β and γ are matrices with coefficients to adequately describe the linear dependency of log(λ) and logit(p) on B and G respectively. If p and λ depend on the same explanatory variables, the number of model parameters may be reduced by expressing p as a function of λ. Lambert pro- poses the relation logit(p) = −τ log(λ) for some τ 2 R. This implies that 1 pi = . 1+λi The resulting model is denoted by ZIP (τ). 8 3. Models for Semicontinuous Data with Excess Zeros There are a number of different models which can be applied to semicontinuous data with excess zeros. We will present a few of the most common ones. In all of these models we will let Y denote the observed variable. The models we will present are all special kinds of two-component mixture models. A mixture model with two components has the form Y = ∆Z1 + (1 − ∆)Z2 Z1 Z2 with ∆ ∼ Ber(p), Z1 ∼ P and Z2 ∼ P . Z2 In the models we will present, we have that P = δf0g. 3.1 Tobit Models The Tobit model which was proposed by James Tobin in 1958 [03] assumes that Y can be expressed in terms of a latent variable Y ∗ which can only be observed for values greater than zero.