Count Models 1 Sociology 8811 Lecture 12 Copyright © 2007 by Evan Schofer Do not copy or distribute without permission

Count Variables Many dependent variables are counts: Non-negative integers # Crimes a person has committed in lifetime # Children living in a household # new companies founded in a year (in an industry) # of social protests per month in a city Can you think of others?

Count Variables Count variables can be modeled with OLS regression… but: 1. Linear models can yield negative predicted values… whereas counts are never negative Similar to the problem of the Linear Probability Model 2. Count variables are often highly skewed Ex: # crimes committed this year… most people are zero or very low; a few people are very high Extreme skew violates the normality assumption of OLS regression.

Count Models Two most common count models: Poisson Regression Model Negative Binomial Regression Model Both based on the Poisson distribution: m = expected count (and variance) Called lambda (l) in some texts; I rely on Freese & Long 2006 y = observed count

Poisson Regression Strategy: Model log of m as a function of Xs Quite similar to modeling log odds in logit Again, the log form avoids negative values Which can be written as:

Poisson Regression: Example Hours per week spent on web

Poisson Regression: Web Use Output = similar to logistic regression . poisson wwwhr male age educ lowincome babies Poisson regression Number of obs = 1552 LR chi2(5) = 525.66 Prob > chi2 = 0.0000 Log likelihood = -8598.488 Pseudo R2 = 0.0297 ------------------------------------------------------------------------------ wwwhr | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male | .3595968 .0210578 17.08 0.000 .3183242 .4008694 age | -.0097401 .0007891 -12.34 0.000 -.0112867 -.0081934 educ | .0205217 .004046 5.07 0.000 .0125917 .0284516 lowincome | -.1168778 .0236503 -4.94 0.000 -.1632316 -.0705241 babies | -.1436266 .0224814 -6.39 0.000 -.1876892 -.0995639 _cons | 1.806489 .0641575 28.16 0.000 1.680743 1.932236 Men spend more time on the web than women Number of young children in household reduces web use

Poisson Regression: Stata Output Stata output yields familiar statistics: Standard errors, z/t- values, and p-values for coefficient hypothesis tests Pseudo R-square for model fit Not a great measure… but gives a crude explained variance MLE log likelihood Likelihood ratio test: Chi-square and p-value Comparing to null model (constant only) Tests can also be conducted on nested models with stata command “lrtest”.

Interpreting Coefficients In Poisson Regression, Y is typically conceptualized as a rate… Positive coefficients indicate higher rate; negative = lower rate Like logit, Poisson models are non-linear Coefficients don’t have a simple linear interpretation Like logit, model has a log form; exponentiation aids interpretation Exponentiated coefficients are multiplicative Analogous to odds ratios… but called “incidence rate ratios”.

Interpreting Coefficients Exponentiated coefficients: indicate effect of unit change of X on rate In STATA: “incidence rate ratios”: “poison … , irr” eb= 2.0 indicates that the rate doubles for each unit change in X eb= .5 indicates that the rate drops by half for each unit change in X Recall: Exponentiated coefs are multiplicative If eb= 5.0, a 2-point change in X isn’t 10; it is 5 * 5 = 25 Also: you must invert to see opposite effects If eb= 5.0, a 1-point decrease in X isn’t -5, it is 1/5 = .2

Interpreting Coefficients Again, exponentiated coefficients (rate ratios) can be converted to % change Formula: (eb - 1) * 100% Ex: (e.5 - 1) * 100% = 50% decrease in rate.

Interpreting Coefficients Exponentiated coefficients yield multiplier: . poisson wwwhr male age educ lowincome babies Poisson regression Number of obs = 1552 LR chi2(5) = 525.66 Prob > chi2 = 0.0000 Log likelihood = -8598.488 Pseudo R2 = 0.0297 ------------------------------------------------------------------------------ wwwhr | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male | .3595968 .0210578 17.08 0.000 .3183242 .4008694 age | -.0097401 .0007891 -12.34 0.000 -.0112867 -.0081934 educ | .0205217 .004046 5.07 0.000 .0125917 .0284516 lowincome | -.1168778 .0236503 -4.94 0.000 -.1632316 -.0705241 babies | -.1436266 .0224814 -6.39 0.000 -.1876892 -.0995639 _cons | 1.806489 .0641575 28.16 0.000 1.680743 1.932236 Exponentiation of .359 = 1.43; Rate is 1.43 times higher for men (1.43-1) * 100 = 43% more Exp(-.14) = .87. Each baby reduces rate by factor of .87 (.87-1) * 100 = 13% less

Predicted Counts Stata “predict varname, n” computes predicted value for each case . predict predwww if e(sample), n . list wwwhr predwww if e(sample) +------------------+ | wwwhr predwww | |------------------| 1. | 1 5.659943 | 2. | 3 7.090338 | 3. | 2 5.281404 | 12. | 5 6.09473 | 13. | 4 6.968055 | 15. | 3 5.815624 | 16. | 0 5.539187 | 19. | 0 7.207257 | 20. | 8 8.03906 | 21. | 5 4.400002 | 23. | 1 6.77004 | 24. | 1 4.806245 | 25. | 8 5.710855 | 27. | 12 3.687142 | 33. | 40 4.997193 | Some of the predictions are close to the observed values… Many of the predictions are quite bad… Recall that the model fit was VERY poor!

Predicted Probabilities Stata extension “prcount” can compute probabilities for each possible count outcome For all cases, of for particular groups It plugs values (m), Xs, & bs into formula: Rate: 5.7446 [ 5.6238, 5.8655] Pr(y=0|x): 0.0032 [ 0.0028, 0.0036] Pr(y=1|x): 0.0184 [ 0.0165, 0.0202] Pr(y=2|x): 0.0528 [ 0.0486, 0.0570] Pr(y=3|x): 0.1011 [ 0.0953, 0.1069] Pr(y=4|x): 0.1452 [ 0.1399, 0.1505] Pr(y=5|x): 0.1668 [ 0.1642, 0.1694] Pr(y=6|x): 0.1597 [ 0.1589, 0.1606] Pr(y=7|x): 0.1311 [ 0.1276, 0.1345] Pr(y=8|x): 0.0941 [ 0.0897, 0.0986] Pr(y=9|x): 0.0601 [ 0.0560, 0.0642] male age educ lowincome babies x= .4503866 40.992912 14.345361 .7371134 .20296392

Issue: Exposure Poisson outcome variables are typically conceptualized as rates Web hours per week Number of crimes committed in past year Issue: Cases may vary in exposure to “risk” of a given outcome To properly model rates, we must account for the fact that some cases have greater exposure than others Ex: # crimes committed in lifetime Older people have greater opportunity to have higher counts Alternately, exposure may vary due to research design Ex: Some cases followed for longer time than others…

Issue: Exposure Poisson (and other count models) can address varying exposure: Where ti = exposure time for case i It is easy to incorporate into stata, too: Ex: poisson NumCrimes SES income, exposure(age) Note: Also works with other “count” models.

Poisson Model Assumptions Poisson regression makes a big assumption: That variance of m = m (“equidisperson”) In other words, the mean and variance are the same This assumption is often not met in real data Dispersion is often greater than m: overdispersion Consequence of overdispersion: Standard errors will be underestimated Potential for overconfidence in results; rejecting H0 when you shouldn’t! Note: overdispersion doesn’t necessarily affect predicted counts (compared to alternative models).

Poisson Model Assumptions Overdispersion is most often caused by highly skewed dependent variables Often due to variables with high numbers of zeros Ex: Number of traffic tickets per year Most people have zero, some can have 50! Mean of variable is low, but SD is high Other examples of skewed outcomes # of scholarly publications # cigarettes smoked per day # riots per year (for sample of cities in US).

Negative Binomial Regression Strategy: Modify the Poisson model to address overdispersion Add an “error” term to the basic model: Additional model assumptions: Expected value of exponentiated error = 1 (ee = 1) Exponentiated error is Gamma distributed We hope that these assumptions are more plausible than the equidispersion assumption!

Negative Binomial Regression Full negative biniomial model: Note that the model incorporates a new parameter: a Alpha represents the extent of overdispersion If a = 0 the model reduces to simple poisson regression

Negative Binomial Regression Question: Is alpha (a) = 0? If so, we can use Poisson regression If not, overdispersion is present; Poisson is inadequate Strategy: conduct a statistical test of the hypothesis: H0: a = 0; H1: a > 0 Stata provides this information when you run a negative binomial model: Likelihood ratio test (G2) for alpha P-value < .05 indicates that overdispersion is present; negative binomial is preferred If P>.05, just use Poisson regression So you don’t have to make assumptions about gamma dist….

Negative Binomial Regression Interpreting coefficients: Identical to poisson regression Predicted probabilities: Can be done. You must use big Neg Binomial formula Plugging in observed Xs, estimates of a, Bs… Probably best to get STATA to do this one… Long & Freese created command: prvalue

Negative Binomial Example: Web Use Note: Bs are similar but SEs change a lot! Negative binomial regression Number of obs = 1552 LR chi2(5) = 57.80 Prob > chi2 = 0.0000 Log likelihood = -4368.6846 Pseudo R2 = 0.0066 ------------------------------------------------------------------------------ wwwhr | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male | .3617049 .0634391 5.70 0.000 .2373666 .4860433 age | -.0109788 .0024167 -4.54 0.000 -.0157155 -.006242 educ | .0171875 .0120853 1.42 0.155 -.0064992 .0408742 lowincome | -.0916297 .0724074 -1.27 0.206 -.2335457 .0502862 babies | -.1238295 .0624742 -1.98 0.047 -.2462767 -.0013824 _cons | 1.881168 .1966654 9.57 0.000 1.495711 2.266625 /lnalpha | .2979718 .0408267 .217953 .3779907 alpha | 1.347124 .0549986 1.243529 1.459349 Likelihood-ratio test of alpha=0: chibar2(01) = 8459.61 Prob>=chibar2 = 0.000 Note: Standard Error for education increased from .004 to .012! Effect is no longer statistically significant.

Negative Binomial Example: Web Use Note: Info on overdispersion is provided Negative binomial regression Number of obs = 1552 LR chi2(5) = 57.80 Prob > chi2 = 0.0000 Log likelihood = -4368.6846 Pseudo R2 = 0.0066 ------------------------------------------------------------------------------ wwwhr | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male | .3617049 .0634391 5.70 0.000 .2373666 .4860433 age | -.0109788 .0024167 -4.54 0.000 -.0157155 -.006242 educ | .0171875 .0120853 1.42 0.155 -.0064992 .0408742 lowincome | -.0916297 .0724074 -1.27 0.206 -.2335457 .0502862 babies | -.1238295 .0624742 -1.98 0.047 -.2462767 -.0013824 _cons | 1.881168 .1966654 9.57 0.000 1.495711 2.266625 /lnalpha | .2979718 .0408267 .217953 .3779907 alpha | 1.347124 .0549986 1.243529 1.459349 Likelihood-ratio test of alpha=0: chibar2(01) = 8459.61 Prob>=chibar2 = 0.000 Alpha is clearly > 0! Overdispersion is evident; LR test p<.05 You should not use Poisson Regression in this case

General Remarks Poisson & Negative binomial models suffer all the same basic issues as “normal” regression Model specification / omitted variable bias Multicollinearity Outliers/influential cases Also, it uses Maximum Likelihood N > 500 = fine; N < 100 can be worrisome Results aren’t necessarily wrong if N<100; But it is a possibility; and hard to know when problems crop up Plus ~10 cases per independent variable.

General Remarks It is often useful to try both Poisson and Negative Binomial models The latter allows you to test for overdispersion Use LRtest on alpha (a) to guide model choice If you don’t suspect dispersion and alpha appears to be zero, use Poission Regression It makes fewer assumptions Such as gamma-distributed error.

Example: Labor Militancy Isaac & Christiansen 2002 Note: Results are presented as % change