1 Discrete and Categorical Data William N. Evans Department of Economics University of Maryland

2 Part I Introduction

3 Workhorse statistical model in social sciences is the multivariate regression model Ordinary least squares (OLS) y i = β 0 + x 1i β 1 + x 2i β 2 +… x ki β k + ε i y i = x i β + ε i

4 Linear model y i =  +  x i +  i  and  are “population” values – represent the true relationship between x and y Unfortunately – these values are unknown The job of the researcher is to estimate these values Notice that if we differentiate y with respect to x, we obtain dy/dx = 

5  represents how much y will change for a fixed change in x –Increase in income for more education –Change in crime or bankruptcy when slots are legalized –Increase in test score if you study more

6 Put some concreteness on the problem State of Maryland budget problems –Drop in revenues –Expensive k-12 school spending initiatives Short-term solution – raise tax on cigarettes by 34 cents/pack Problem – a tax hike will reduce consumption of taxable product Question for state – as taxes are raised, how much will cigarette consumption fall?

7 Simple model: y i =  +  x i +  i Suppose y is a state’s per capita consumption of cigarettes x represents taxes on cigarettes Question – how much will y fall if x is increased by 34 cents/pack? Problem – many reasons why people smoke – cost is but one of them –

8 Data –(Y) State per capita cigarette consumption for the years –(X) tax (State + Federal) in real cents per pack –“Scatter plot” of the data –Negative covariance between variables When x>, more likely that y<  When x  Goal: pick values of  and  that “best fit” the data –Define best fit in a moment

9 Notation True model y i =  +  x i +  i We observe data points (y i,x i ) The parameters  and  are unknown The actual error (  i ) is unknown Estimated model (a,b) are estimates for the parameters ( ,  ) e i is an estimate of  i where e i =y i -a-bx i How do you estimate a and b?

10 Objective: Minimize sum of squared errors Min  i e i 2 =  i (y i – a – bx i ) 2 Minimize the sum of squared errors (SSE) Treat positive and negative errors equally –Over or under predict by “5” is the same magnitude of error –“Quadratic form” –The optimal value for a and b are those that make the 1 st derivative equal zero –Functions reach min or max values when derivatives are zero

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13 The model has a lot of nice features –Statistical properties easy to establish –Optimal estimates easy to obtain –Parameter estimates are easy to interpret –Model maximizes prediction If you minimize SSE you maximize R 2 The model does well as a first order approximation to lots of problems

14 Discrete and Qualitative Data The OLS model work well when y is a continuous variable –Income, wages, test scores, weight, GDP Does not has as many nice properties when y is not continuous Example: doctor visits Integer values Low counts for most people Mass of observations at zero

15 Downside of forcing non-standard outcomes into OLS world? Can predict outside the allowable range –e.g., negative MD visits Does not describe the data generating process well –e.g., mass of observations at zero Violates many properties of OLS –e.g. heteroskedasticity

16 This talk Look at situations when the data generating process does lend itself well to OLS models Mathematically describe the data generating process Show how we use different optimization procedure to obtain estimates Describe the statistical properties

17 Show how to interpret parameters Illustrate how to estimate the models with popular program STATA

18 Types of data generating processes we will consider Dichotomous events (yes or no) –1=yes, 0=no –Graduate high school? work? Are obese? Smoke? Ordinal data –Self reported health (fair, poor, good, excel) –Strongly disagree, disagree, agree, strongly agree

19 Count data –Doctor visits, lost workdays, fatality counts Duration data –Time to failure, time to death, time to re- employment

20 Econometric Resources Recommended textbook –Jeffrey Wooldridge, undergraduate and grad –Lots of insight and mathematical/statistical detail –Very good examples Helpful web sites –My graduate class –Jeff Smith’s class

21 Part II A quick introduction to STATA

22 STATA Very fast, convenient, well-documented, cheap and flexible statistical package Excellent for cross-section/panel data projects, not as great for time series but getting better Not as easy to manipulate large data sets from flat files as SAS I usually clean data in SAS, estimate models in STATA

23 Key characteristic of STATA –All data must be loaded into RAM –Computations are very fast –But, size of the project is limited by available memory Results can be generated two different ways –Command line –Write a program, (*.do) then submit from the command line

24 Sample program to get you started cps87_or.do Program gets you to the point where can Load data into memory Construct new variables Get simple statistics Run a basic regression Store the results on a disk

25 Data (cps87_do.dta) Random sample of data from 1987 Current Population Survey outgoing rotation group Sample selection –Males –21-64 –Working 30+hours/week 19,906 observations

26 Major caveat Hardest thing to learn/do: get data from some other source and get it into STATA data set We skip over that part All the data sets are loaded into a STATA data file that can be called by saying: use data file name

27 Housekeeping at the top of the program * this line defines the semicolon as the ; * end of line delimiter; # delimit ; * set memork for 10 meg; set memory 10m; * write results to a log file; * the replace options writes over old; * log files; log using cps87_or.log,replace; * open stata data set; use c:\bill\stata\cps87_or; * list variables and labels in data set; desc;

> - storage display value variable name type format label variable label > - age float %9.0g age in years race float %9.0g 1=white, non-hisp, 2=place, n.h, 3=hisp educ float %9.0g years of education unionm float %9.0g 1=union member, 2=otherwise smsa float %9.0g 1=live in 19 largest smsa, 2=other smsa, 3=non smsa region float %9.0g 1=east, 2=midwest, 3=south, 4=west earnwke float %9.0g usual weekly earnings

29 Constructing new variables Use ‘gen’ command for generate new variables Syntax –gen new variable name=math statement Easily construct new variables via –Algebraic operations –Math/trig functions (ln, exp, etc.) –Logical operators (when true, =1, when false, =0)

30 From program * generate new variables; * lines 1-2 illustrate basic math functoins; * lines 3-4 line illustrate logical operators; * line 5 illustrate the OR statement; * line 6 illustrates the AND statement; * after you construct new variables, compress the data again; gen age2=age*age; gen earnwkl=ln(earnwke); gen union=unionm==1; gen topcode=earnwke==999; gen nonwhite=((race==2)|(race==3)); gen big_ne=((region==1)&(smsa==1));

31 Getting basic statistics desc -- describes variables in the data set sum – gets summary statistics tab – produces frequencies (tables) of discrete variables

32 * get descriptive statistics; sum; * get detailed descriptics for continuous variables; sum earnwke, detail; * get frequencies of discrete variables; tabulate unionm; tabulate race; * get two-way table of frequencies; tabulate region smsa, row column cell;

33 STATA Resources - Specific “Regression Models for Categorical Dependent Variables Using STATA” –J. Scott Long and Jeremy Freese Available for sale from STATA website for $52 ( Post-estimation subroutines that translate results –Do not need to buy the book to use the subroutines

34 In STATA command line type net search spost Will give you a list of available programs to download One is Spostado from Click on the link and install the files

35 Continuous Distributions Random variables with infinite number of possible values Examples -- units of measure (time, weight, distance) Many discrete outcomes can be treated as continuous, e.g., SAT scores

36 How to describe a continuous random variable The Probability Density Function (PDF) The PDF for a random variable x is defined as f(x), where f(x) $ 0 I f(x)dx = 1 Calculus review: The integral of a function gives the “area under the curve”

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38 Cumulative Distribution Function (CDF) Suppose x is a “measure” like distance or time 0 # x # 4 We may be interested in the Pr(x # a) ?

39 CDF What if we consider all values?

40 Properties of CDF Note that Pr(x # b) + Pr(x>b) =1 Pr(x>b) = 1 – Pr(x # b) Many times, it is easier to work with compliments

41 General notation for continuous distributions The PDF is described by lower case such as f(x) The CDF is defined as upper case such as F(a)

42 Standard Normal Distribution Most frequently used continuous distribution Symmetric “bell-shaped” distribution As we will show, the normal has useful properties Many variables we observe in the real world look normally distributed. Can translate normal into ‘standard normal’

43 Examples of variables that look normally distributed IQ scores SAT scores Heights of females Log income Average gestation (weeks of pregnancy) As we will show in a few weeks – sample means are normally distributed!!!

44 Standard Normal Distribution PDF: For -  # z # 

45 Notation  (z) is the standard normal PDF evaluated at z  [a] = Pr(z  a)

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47 Standard Normal Notice that: –Normal is symmetric:  (a) =  (-a) –Normal is “unimodal” –Median=mean –Area under curve=1 –Almost all area is between (-3,3) Evaluations of the CDF are done with –Statistical functions (excel, SAS, etc) –Tables

48 Standard Normal CDF Pr(z  -0.98) =  [-0.98] =

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50 Pr(z  1.41) =  [1.41] =

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52 Pr(x>1.17) = 1 – Pr(z  1.17) = 1-  [1.17] = 1 – =

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54 Pr(0.1  z  1.9) = Pr(z  1.9) – Pr(z  0.1) = M (1.9) - M (0.1) = =

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58 Important Properties of Normal Distribution Pr(z  A) =  [A] Pr(z > A) = 1 -  [A] Pr(z  - A) =  [-A] Pr(z > -A) = 1 -  [-A] =  [A]

59 Maximum likelihood estimation Observe n independent outcomes, all drawn from the same distribution (y 1, y 2, y 3 ….y n ) y i is drawn from f(y i ; θ) where θ is an unknown parameter for the distribution f Recall definition of indepedence. If a and b and independent, Prob(a and b) = Pr(a)Pr(B)

60 Because all the draws are independent, the probability these particular n values of Y would be drawn at random is called the ‘likelihood function’ and it equals L = Pr(y 1 )Pr(y 2 )…Pr(y n ) L = f(y 1 ; θ)f(y 2 ; θ)…..f(y 3 ; θ)

61 MLE: pick a value for θ that best represents the chance these n values of y would have been generated randomly To maximize L, maximize a monotonic function of L Recall ln(abcd)=ln(a)+ln(b)+ln(c)+ln(d)

62 Max L = ln(L) = ln[f(y 1 ; θ)] +ln[f(y 2 ; θ)] + ….. ln[f(y n ; θ) = Σ i ln[f(y i ; θ)] Pick θ so that L is maximized d L /dθ = 0

63 L θ θ1θ1 θ2θ2

64 Example: Poisson Suppose y measures ‘counts’ such as doctor visits. y i is drawn from a Poisson distribution f(y i ;λ) =e -λ λ y i /y i ! For λ>0 E[y i ]= Var[y i ] = λ

65 Given n observations, (y 1, y 2, y 3 ….y n ) Pick value of λ that maximizes L Max L = Σ i ln[f(y i ; θ)] = Σ i ln[e -λ λ y i /y i !] = Σ i [– λ + y i ln(λ) – ln(y i !)] = -n λ + ln(λ) Σ i y i – Σ i ln(y i !)

66 L = -n λ + ln(λ) Σ i y i – Σ i ln(y i !) d L /dθ = -n + (1/ λ )Σ i y i = 0 Solve for λ λ = Σ i y i /n =  = sample mean of y

67 In most cases however, cannot find a ‘closed form’ solution for the parameter in ln[f(y i ; θ)] Must ‘search’ over all possible solutions How does the search work? Start with candidate value of θ. Calculate d L /dθ

68 If d L /dθ > 0, increasing θ will increase L so we increase θ some If d L /dθ < 0, decreasing θ will increase L so we decrease θ some Keep changing θ until d L /dθ = 0 How far you ‘step’ when you change θ is determined by a number of different factors

69 L θθ1θ1 d L/d θ > 0

70 L θ θ3θ3 d L/d θ < 0

71 Properties of MLE estimates Sometimes call efficient estimation. Can never generate a smaller variance than one obtained by MLE Parameters estimates are distributed as a normal distribution when samples sizes are large

72 Section 3 Probit and Logit Models

73 Dichotomous Data Suppose data is discrete but there are only 2 outcomes Examples –Graduate high school or not –Patient dies or not –Working or not –Smoker or not In data, y i =1 if yes, y i =0 if no

74 How to model the data generating process? There are only two outcomes Research question: What factors impact whether the event occurs? To answer, will model the probability the outcome occurs Pr(Y i =1) when y i =1 or Pr(Y i =0) = 1- Pr(Y i =1) when y i =0

75 Think of the problem from a MLE perspective Likelihood for i’th observation L i = Pr(Y i =1) Yi [1 - Pr(Y i =1)] (1-Yi) When y i =1, only relevant part is Pr(Y i =1) When y i =0, only relevant part is [1 - Pr(Y i =1)]

76 L = Σ i ln[L i ] = = Σ i {y i ln[Pr(y i =1)] + (1-y i )ln[Pr(y i =0)] } Notice that up to this point, the model is generic. The log likelihood function will determined by the assumptions concerning how we determine Pr(y i =1)

77 Modeling the probability There is some process (biological, social, decision theoretic, etc) that determines the outcome y Some of the variables impacting are observed, some are not Requires that we model how these factors impact the probabilities Model from a ‘latent variable’ perspective

78 Consider a women’s decision to work y i * = the person’s net benefit to work Two components of y i * –Characteristics that we can measure Education, age, income of spouse, prices of child care –Some we cannot measure How much you like spending time with your kids how much you like/hate your job

79 We aggregate these two components into one equation y i * = β 0 + x 1i β 1 + x 2i β 2 +… x ki β k + ε i = x i β + ε i x i β (measurable characteristics but with uncertain weights) ε i random unmeasured characteristics Decision rule: person will work if y i * > 0 (if net benefits are positive) y i =1 if y i *>0 y i =0 if y i * ≤0

80 y i =1 if y i *>0 y i * = x i β + ε i > 0 only if ε i > - x i β y i =0 if y i * ≤0 y i * = x i β + ε i ≤ 0 only if ε i ≤ - x i β

81 Suppose x i β is ‘big.’ –High wages –Low husband’s income –Low cost of child care We would expect this person to work, UNLESS, there is some unmeasured ‘variable’ that counteracts this

82 Suppose a mom really likes spending time with her kids, or she hates her job. The unmeasured benefit of working has a big negative coefficient ε i If we observe them working, ε i must not have been too big, since y i =1 if ε i > - x i β

83 Consider the opposite. Suppose we observe someone NOT working. Then ε i must not have been big, since y i =0 if ε i ≤ - x i β

84 Logit Recall y i =1 if ε i > - x i β Since ε i is a logistic distribution Pr(ε i > - x i β) = 1 – F(- x i β) The logistic is also a symmetric distribution, so 1 – F(- x i β) = F(x i β) = exp(x i β)/(1+exp(x i β))

85 When ε i is a logistic distribution Pr(y i =1) = exp(x i β)/(1+exp(x i β)) Pr(y i =0) = 1/(1+exp(x i β))

86 Example: Workplace smoking bans Smoking supplements to 1991 and 1993 National Health Interview Survey Asked all respondents whether they currently smoke Asked workers about workplace tobacco policies Sample: workers Key variables: current smoking and whether they faced by workplace ban

87 Data: workplace1.dta Sample program: workplace1.doc Results: workplace1.log

88 Description of variables in data. desc; storage display value variable name type format label variable label > - smoker byte %9.0g is current smoking worka byte %9.0g has workplace smoking bans age byte %9.0g age in years male byte %9.0g male black byte %9.0g black hispanic byte %9.0g hispanic incomel float %9.0g log income hsgrad byte %9.0g is hs graduate somecol byte %9.0g has some college college float %9.0g

89 Summary statistics sum; Variable | Obs Mean Std. Dev. Min Max smoker | worka | age | male | black | hispanic | incomel | hsgrad | somecol | college |

90 Running a probit probit smoker age incomel male black hispanic hsgrad somecol college worka; The first variable after ‘probit’ is the discrete outcome, the rest of the variables are the independent variables Includes a constant as a default

91 Running a logit logit smoker age incomel male black hispanic hsgrad somecol college worka; Same as probit, just change the first word

92 Running linear probability reg smoker age incomel male black hispanic hsgrad somecol college worka, robust; Simple regression. Standard errors are incorrect (heteroskedasticity) robust option produces standard errors with arbitrary form of heteroskedasticity

93 Probit Results Probit estimates Number of obs = LR chi2(9) = Prob > chi2 = Log likelihood = Pseudo R2 = smoker | Coef. Std. Err. z P>|z| [95% Conf. Interval] age | incomel | male | black | hispanic | hsgrad | somecol | college | worka | _cons |

94 How to measure fit? Regression (OLS) –minimize sum of squared errors –Or, maximize R 2 –The model is designed to maximize predictive capacity Not the case with Probit/Logit –MLE models pick distribution parameters so as best describe the data generating process –May or may not ‘predict’ the outcome well

95 Pseudo R 2 LL k log likelihood with all variables LL 1 log likelihood with only a constant 0 > LL k > LL 1 so | LL k | < |LL 1 | Pseudo R 2 = 1 - |LL 1 /LL k | Bounded between 0-1 Not anything like an R 2 from a regression

96 Predicting Y Let b be the estimated value of β For any candidate vector of x i, we can predict probabilities, P i P i = Ф(x i b) Once you have P i, pick a threshold value, T, so that you predict Y p = 1 if P i > T Y p = 0 if P i ≤ T Then compare, fraction correctly predicted

97 Question: what value to pick for T? Can pick.5 –Intuitive. More likely to engage in the activity than to not engage in it –However, when the  is small, this criteria does a poor job of predicting Y i =1 –However, when the  is close to 1, this criteria does a poor job of picking Y i =0

98 *predict probability of smoking; predict pred_prob_smoke; * get detailed descriptive data about predicted prob; sum pred_prob, detail; * predict binary outcome with 50% cutoff; gen pred_smoke1=pred_prob_smoke>=.5; label variable pred_smoke1 "predicted smoking, 50% cutoff"; * compare actual values; tab smoker pred_smoke1, row col cell;

99. sum pred_prob, detail; Pr(smoker) Percentiles Smallest 1% % % Obs % Sum of Wgt % Mean Largest Std. Dev % % Variance % Skewness % Kurtosis

100 Notice two things –Sample mean of the predicted probabilities is close to the sample mean outcome –99% of the probabilities are less than.5 –Should predict few smokers if use a 50% cutoff

101 | predicted smoking, is current | 50% cutoff smoking | 0 1 | Total | 12, | 12,167 | | | | | | | 4, | 4,091 | | | | | | Total | 16, | 16,258 | | | | | |

102 Check on-diagonal elements. The last number in each 2x2 element is the fraction in the cell The model correctly predicts = 74.90% of the obs It only predicts a small fraction of smokers

103 Do not be amazed by the 75% percent correct prediction If you said everyone has a  chance of smoking (a case of no covariates), you would be correct Max[( ,(1-  )] percent of the time

104 In this case, 25.16% smoke. If everyone had the same chance of smoking, we would assign everyone Pr(y=1) =.2516 We would be correct for the = people who do not smoke

105 Key points about prediction MLE models are not designed to maximize prediction Should not be surprised they do not predict well In this case, not particularly good measures of predictive capacity

106 Translating coefficients in probit: Continuous Covariates Pr(y i =1) = Φ[β 0 + x 1i β 1 + x 2i β 2 +… x ki β k ] Suppose that x 1i is a continuous variable d Pr(y i =1) /d x 1i = ? What is the change in the probability of an event give a change in x 1i?

107 Marginal Effect d Pr(y i =1) /d x 1i = β 1 φ[β 0 + x 1i β 1 + x 2i β 2 +… x ki β k ] Notice two things. Marginal effect is a function of the other parameters and the values of x.

108 Translating Coefficients: Discrete Covariates Pr(y i =1) = Φ[β 0 + x 1i β 1 + x 2i β 2 +… x ki β k ] Suppose that x 2i is a dummy variable (1 if yes, 0 if no) Marginal effect makes no sense, cannot change x 2i by a little amount. It is either 1 or 0. Redefine the variable of interest. Compare outcomes with and without x 2i

109 y 1 = Pr(y i =1 | x 2i =1) = Φ[β 0 + x 1i β 1 + β 2 + x 3i β 3 +… ] y 0 = Pr(y i =1 | x 2i =0) = Φ[β 0 + x 1i β 1 + x 3i β 3 … ] Marginal effect = y 1 – y 0. Difference in probabilities with and without x 2i?

110 In STATA Marginal effects for continuous variables, STATA picks sample means for X’s Change in probabilities for dichotomous outcomes, STATA picks sample means for X’s

111 STATA command for Marginal Effects mfx compute; Must be after the outcome when estimates are still active in program.

112 Marginal effects after probit y = Pr(smoker) (predict) = variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X age | incomel | male*| black*| hispanic*| hsgrad*| somecol*| college*| worka*| (*) dy/dx is for discrete change of dummy variable from 0 to 1

113 Interpret results 10% increase in income will reduce smoking by 2.9 percentage points 10 year increase in age will decrease smoking rates.4 percentage points Those with a college degree are 21.5 percentage points less likely to smoke Those that face a workplace smoking ban have 6.7 percentage point lower probability of smoking

114 Do not confuse percentage point and percent differences –A 6.7 percentage point drop is 29% of the sample mean of 24 percent. –Blacks have smoking rates that are 3.2 percentage points lower than others, which is 13 percent of the sample mean

115 Comparing Marginal Effects VariableLPProbitLogit age incomel male Black hispanic hsgrad college worka

116 When will results differ? Normal and logit CDF look –Similar in the mid point of the distribution –Different in the tails You obtain more observations in the tails of the distribution when –Samples sizes are large –  approaches 1 or 0 These situations will produce more differences in estimates

117 Some nice properties of the Logit Outcome, y=1 or 0 Treatment, x=1 or 0 Other covariates, x Context, –x = whether a baby is born with a low weight birth –x = whether the mom smoked or not during pregnancy

118 Risk ratio RR = Prob(y=1|x=1)/Prob(y=1|x=0) Differences in the probability of an event when x is and is not observed How much does smoking elevate the chance your child will be a low weight birth

119 Let Y yx be the probability y=1 or 0 given x=1 or 0 Think of the risk ratio the following way Y 11 is the probability Y=1 when X=1 Y 10 is the probability Y=1 when X=0 Y 11 = RR*Y 10

120 Odds Ratio OR=A/B = [Y 11 /Y 01 ]/[Y 10 /Y 00 ] A = [Pr(Y=1|X=1)/Pr(Y=0|X=1)] = odds of Y occurring if you are a smoker B = [Pr(Y=1|X=0)/Pr(Y=0|X=0)] = odds of y happening if you are not a smoker What are the relative odds of Y happening if you do or do not experience X

121 Suppose Pr(Y i =1) = F(β o + β 1 X i + β 2 Z) and F is the logistic function Can show that OR = exp(β 1 ) = e β1 This number is typically reported by most statistical packages

122 Details Y 11 = exp(β o + β 1 + β 2 Z) /(1+ exp(β o + β 1 + β 2 Z) ) Y 10 = exp(β o + β 2 Z)/(1+ exp(β o +β 2 Z)) Y 01 = 1 /(1+ exp(β o + β 1 + β 2 Z) ) Y 00 = 1/(1+ exp(β o +β 2 Z) [Y 11 /Y 01 ] = exp(β o + β 1 + β 2 Z) [Y 10 /Y 00 ] = exp(β o + β 2 Z) OR=A/B = [Y 11 /Y 01 ]/[Y 10 /Y 00 ] = exp(β o + β 1 + β 2 Z)/ exp(β o + β 2 Z) = exp(β 1 )

123 Suppose Y is rare,  close to 0 –Pr(Y=0|X=1) and Pr(Y=0|X=0) are both close to 1, so they cancel Therefore, when  is close to 0 –Odds Ratio = Risk Ratio Why is this nice?

124 Population attributable risk Average outcome in the population  = (1- ) Y 10 + Y 11 = (1- )Y 10 + (RR)Y 10 Average outcomes are a weighted average of outcomes for X=0 and X=1 What would the average outcome be in the absence of X (e.g., reduce smoking rates to 0) Y a = Y 10

125 Population Attributable Risk PAR Fraction of outcome attributed to X The difference between the current rate and the rate that would exist without X, divided by the current rate PAR = (  – Y a )/  = (RR – 1) /[(1- ) + RR ]

126 Example: Maternal Smoking and Low Weight Births 6% births are low weight –< 2500 grams ( –Average birth is 3300 grams (5.5 lbs) Maternal smoking during pregnancy has been identified as a key cofactor –13% of mothers smoke –This number was falling about 1 percentage point per year during 1980s/90s –Doubles chance of low weight birth

127 Natality detail data Census of all births (4 million/year) Annual files starting in the 60s Information about –Baby (birth weight, length, date, sex, plurality, birth injuries) –Demographics (age, race, marital, educ of mom) –Birth (who delivered, method of delivery) –Health of mom (smoke/drank during preg, weight gain)

128 Smoking not available from CA or NY ~3 million usable observations I pulled.5% random sample from 1995 About 12,500 obs Variables: birthweight (grams), smoked, married, 4-level race, 5 level education, mothers age at birth

> - storage display value variable name type format label variable label > - birthw int %9.0g birth weight in grams smoked byte %9.0g =1 if mom smoked during pregnancy age byte %9.0g moms age at birth married byte %9.0g =1 if married race4 byte %9.0g 1=white,2=black,3=asian,4=other educ5 byte %9.0g 1=0-8, 2=9-11, 3=12, 4=13-15, 5=16+ visits byte %9.0g prenatal visits

130 dummy | variable, | =1 | =1 if mom smoked ifBW<2500 | during pregnancy grams | 0 1 | Total | 11,626 1,745 | 13,371 | | | | | | | | 859 | | | | 6.04 | | Total | 12,285 1,945 | 14,230 | | | | | |

131 Notice a few things –13.7% of women smoke –6% have low weight birth Pr(LBW | Smoke) =10.28% Pr(LBW |~ Smoke) = 5.36% RR = Pr(LBW | Smoke)/ Pr(LBW |~ Smoke) = / = 1.92

132 Logit results Log likelihood = Pseudo R2 = lowbw | Coef. Std. Err. z P>|z| [95% Conf. Interval] smoked | age | married | _Ieduc5_2 | _Ieduc5_3 | _Ieduc5_4 | _Ieduc5_5 | _Irace4_2 | _Irace4_3 | _Irace4_4 | _cons |

133 Odds Ratios Smoked –exp(0.674) = 1.96 –Smokers are twice as likely to have a low weight birth _Irace4_2 (Blacks) –exp(0.707) = 2.02 –Blacks are twice as likely to have a low weight birth

134 Asking for odds ratios Logistic y x1 x2; In this case xi: logistic lowbw smoked age married i.educ5 i.race4;

135 Log likelihood = Pseudo R2 = lowbw | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] smoked | age | married | _Ieduc5_2 | _Ieduc5_3 | _Ieduc5_4 | _Ieduc5_5 | _Irace4_2 | _Irace4_3 | _Irace4_4 |

136 PAR PAR = (RR – 1) /[(1- ) + RR ] = RR = 1.96 PAR = % of low weight births attributed to maternal smoking

137 Hypothesis Testing in MLE models MLE are asymptotically normally distributed, one of the properties of MLE Therefore, standard t-tests of hypothesis will work as long as samples are ‘large’ What ‘large’ means is open to question What to do when samples are ‘small’ – table for a moment

138 Testing a linear combination of parameters Suppose you have a probit model Φ[β 0 + x 1i β 1 + x 2i β 2 + x 3i β 3 +… ] Test a linear combination or parameters Simplest example, test a subset are zero β 1 = β 2 = β 3 = β 4 =0 To fix the discussion N observations K parameters J restrictions (count the equals signs, j=4)

139 Wald Test Based on the fact that the parameters are distributed asymptotically normal Probability theory review –Suppose you have m draws from a standard normal distribution (z i ) –M = z z …. Z m 2 –M is distributed as a Chi-square with m degrees of freedom

140 Wald test constructs a ‘quadratic form’ suggested by the test you want to perform This combination, because it contains squares of the true parameters, should, if the hypothesis is true, be distributed as a Chi square with j degrees of freedom. If the test statistic is ‘large’, relative to the degrees of freedom of the test, we reject, because there is a low probability we would have drawn that value at random from the distribution

141 Reading values from a Table All stats books will report the ‘percentiles’ of a chi-square –Vertical axis (degrees of freedom) –Horizontal axis (percentiles) –Entry is the value where ‘percentile’ of the distribution falls below

142 Example: Suppose 4 restrictions 95% of a chi-square distribution falls below So there is only a 5% a number drawn at random will exceed If your test statistic is below, cannot reject null If your test statistics is above, reject null

143 Chi-square

144 Wald test in STATA Default test in MLE models Easy to do. Look at program test hsgrad somecol college Does not estimate the ‘restricted’ model ‘Lower power’ than other tests, i.e., high chance of false negative

Log likelihood test * how to run the same tests with a -2 log like test; * estimate the unresticted model and save the estimates ; * in urmodel; probit smoker age incomel male black hispanic hsgrad somecol college worka; estimates store urmodel; * estimate the restricted model. save results in rmodel; probit smoker age incomel male black hispanic worka; estimates store rmodel; lrtest urmodel rmodel;

146 I prefer -2 log likelihood test –Estimates the restricted and unrestricted model –Therefore, has more power than a Wald test In most cases, they give the same ‘decision’ (reject/not reject)

147 Ordered probit models

148 Ordered Probit Many discrete outcomes are to questions that have a natural ordering but no quantitative interpretation: Examples: –Self reported health status (excellent, very good, good, fair, poor) –Do you agree with the following statement Strongly agree, agree, disagree, strongly disagree

149 Can use the same type of model as in the previous section to analyze these outcomes Another ‘latent variable’ model Key to the model: there is a monotonic ordering of the qualitative responses

150 Self reported health status Excellent, very good, good, fair, poor Coded as 1, 2, 3, 4, 5 on National Health Interview Survey We will code as 5,4,3,2,1 (easier to think of this way) Asked on every major health survey Important predictor of health outcomes, e.g. mortality Key question: what predicts health status?

151 Important to note – the numbers 1-5 mean nothing in terms of their value, just an ordering to show you the lowest to highest The example below is easily adapted to include categorical variables with any number of outcomes

152 Model y i * = latent index of reported health The latent index measures your own scale of health. Once y i * crosses a certain value you report poor, then good, then very good, then excellent health

153 y i = (1,2,3,4,5) for (fair, poor, VG, G, excel) Interval decision rule y i =1 if y i * ≤ u 1 y i =2 if u 1 < y i * ≤ u 2 y i =3 if u 2 < y i * ≤ u 3 y i =4 if u 3 < y i * ≤ u 4 y i =5 if y i * > u 4

154 As with logit and probit models, we will assume y i * is a function of observed and unobserved variables y i * = β 0 + x 1i β 1 + x 2i β 2 …. x ki β k + ε i y i * = x i β + ε i

155 The threshold values (u 1, u 2, u 3, u 4 ) are unknown. We do not know the value of the index necessary to push you from very good to excellent. In theory, the threshold values are different for everyone Computer will not only estimate the β’s, but also the thresholds – average across people

156 As with probit and logit, the model will be determined by the assumed distribution of ε In practice, most people pick nornal, generating an ‘ordered probit’ (I have no idea why) We will generate the math for the probit version

157 Probabilities Lets do the outliers, Pr(y i =1) and Pr(y i =5) first Pr(y i =1) = Pr(y i * ≤ u 1 ) = Pr(x i β +ε i ≤ u 1 ) =Pr(ε i ≤ u 1 - x i β) = Φ[u 1 - x i β] = 1- Φ[x i β – u 1 ]

158 Pr(y i =5) = Pr(y i * > u 4 ) = Pr(x i β +ε i > u 4 ) =Pr(ε i > u 4 - x i β) = 1 - Φ[u 4 - x i β] = Φ[x i β – u 4 ]

159 Sample one for y=3 Pr(y i =3) = Pr(u 2 < y i * ≤ u 3 ) = Pr(y i * ≤ u 3 ) – Pr(y i * ≤ u 2 ) = Pr(x i β +ε i ≤ u 3 ) – Pr(x i β +ε i ≤ u 2 ) = Pr(ε i ≤ u 3 - x i β) - Pr(ε i ≤ u 2 - x i β) = Φ[u 3 - x i β] - Φ[u 2 - x i β] = 1 - Φ[x i β - u 3 ] – 1 + Φ[x i β - u 2 ] = Φ[x i β - u 2 ] - Φ[x i β - u 3 ]

160 Summary Pr(y i =1) = 1- Φ[x i β – u 1 ] Pr(y i =2) = Φ[x i β – u 1 ] - Φ[x i β – u 2 ] Pr(y i =3) = Φ[x i β – u 2 ] - Φ[x i β – u 3 ] Pr(y i =4) = Φ[x i β – u 3 ] - Φ[x i β – u 4 ] Pr(y i =5) = Φ[x i β – u 4 ]

161 Likelihood function There are 5 possible choices for each person Only 1 is observed L = Σ i ln[Pr(y i =k)] for k

162 Programming example Cancer control supplement to 1994 National Health Interview Survey Question: what observed characteristics predict self reported health (1-5 scale) 1=poor, 5=excellent Key covariates: income, education, age, current and former smoking status Programs sr_health_status.do,.dta,.log

163 desc; male byte %9.0g =1 if male age byte %9.0g age in years educ byte %9.0g years of education smoke byte %9.0g current smoker smoke5 byte %9.0g smoked in past 5 years black float %9.0g =1 if respondent is black othrace float %9.0g =1 if other race (white is ref) sr_health float %9.0g 1-5 self reported health, 5=excel, 1=poor famincl float %9.0g log family income

164 tab sr_health; 1-5 self | reported | health, | 5=excel, | 1=poor | Freq. Percent Cum | | | 3, | 3, | 4, Total | 12,

165 In STATA oprobit sr_health male age educ famincl black othrace smoke smoke5;

166 Ordered probit estimates Number of obs = LR chi2(8) = Prob > chi2 = Log likelihood = Pseudo R2 = sr_health | Coef. Std. Err. z P>|z| [95% Conf. Interval] male | age | educ | famincl | black | othrace | smoke | smoke5 | _cut1 | (Ancillary parameters) _cut2 | _cut3 | _cut4 |

167 Interpret coefficients Marginal effects/changes in probabilities are now a function of 2 things –Point of expansion (x’s) –Frame of reference for outcome (y) STATA –Picks mean values for x’s –You pick the value of y

168 Continuous x’s Consider y=5 d Pr(y i =5)/dx i = d Φ[x i β – u 4 ]/dx i = βφ[x i β – u 4 ] Consider y=3 d Pr(y i =3)/dx i = βφ[x i β – u 3 ] - βφ[x i β – u 4 ]

169 Discrete X’s x i β = β 0 + x 1i β 1 + x 2i β 2 …. x ki β k –X 2i is yes or no (1 or 0) ΔPr(y i =5) = Φ[β 0 + x 1i β 1 + β 2 + x 3i β x ki β k ] - Φ[β 0 + x 1i β 1 + x 3i β 3 …. x ki β k ] Change in the probabilities when x 2i =1 and x 2i =0

170 Ask for marginal effects mfx compute, predict(outcome(5));

171 mfx compute, predict(outcome(5)); Marginal effects after oprobit y = Pr(sr_health==5) (predict, outcome(5)) = variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X male*| age | educ | famincl | black*| othrace*| smoke*| smoke5*| (*) dy/dx is for discrete change of dummy variable from 0 to 1

172 Interpret the results Males are 4.7 percentage points more likely to report excellent Each year of age decreases chance of reporting excellent by 0.7 percentage points Current smokers are 7.5 percentage points less likely to report excellent health

173 Minor notes about estimation Wald tests/-2 log likelihood tests are done the exact same was as in PROBIT and LOGIT

174 Use PRCHANGE to calculate marginal effect for a specific person prchange, x(age=40 black=0 othrace=0 smoke=0 smoke5=0 educ=16); –When a variable is NOT specified (famincl), STATA takes the sample mean.

175 PRCHANGE will produce results for all outcomes male Avg|Chg| > >

176 age Avg|Chg| Min->Max / sd/ MargEfct

177 Section Count Data Models

178 Introduction Many outcomes of interest are integer counts –Doctor visits –Low work days –Cigarettes smoked per day –Missed school days OLS models can easily handle some integer models

179 Example –SAT scores are essentially integer values –Few at ‘tails’ –Distribution is fairly continuous –OLS models well In contrast, suppose –High fraction of zeros –Small positive values

180 OLS models will –Predict negative values –Do a poor job of predicting the mass of observations at zero Example –Dr visits in past year, Medicare patients(65+) –1987 National Medical Expenditure Survey –Top code (for now) at 10 –17% have no visits

181 visits | Freq. Percent Cum | | | | | | | | | | | Total | 5,

182 Poisson Model y i is drawn from a Poisson distribution Poisson parameter varies across observations f(y i ;λ i ) =e -λi λ i yi /y i ! For λ i >0 E[y i ]= Var[y i ] = λ i = f(x i, β)

183 λ i must be positive at all times Therefore, we CANNOT let λ i = x i β Let λ i = exp(x i β) ln(λ i ) = (x i β)

184 d ln(λ i )/dx i = β Remember that d ln(λ i ) = dλ i /λ i Interpret β as the percentage change in mean outcomes for a change in x

185 Problems with Poisson Variance grows with the mean –E[y i ]= Var[y i ] = λ i = f(x i, β) Most data sets have over dispersion, where the variance grows faster than the mean In dr. visits sample,  = 5.6, s=6.7 Impose Mean=Var, severe restriction and you tend to reduce standard errors

186 Negative Binomial Model Where γ i = exp(x i β) and δ ≥ 0 E[y i ] = δγ i = δexp(x i β) Var[y i ] = δ (1+δ) γ i Var[y i ]/ E[y i ] = (1+δ)

187 δ must always be ≥ 0 In this case, the variance grows faster than the mean If δ=0, the model collapses into the Poisson Always estimate negative binomial If you cannot reject the null that δ=0, report the Poisson estimates

188 Notice that ln(E[y i ]) = ln(δ) + ln(γ i ), so d ln(E[y i ]) /dx i = β Parameters have the same interpretation as in the Poisson model

189 In STATA POISSON estimates a MLE model for poisson –Syntax POISSON y independent variables NBREG estimates MLE negative binomial –Syntax NBREG y independent variables

190 Interpret results for Poisson Those with CHRONIC condition have 50% more mean MD visits Those in EXCELent health have 78% fewer MD visits BLACKS have 33% fewer visits than whites Income elasticity is 0.021, 10% increase in income generates a 2.1% increase in visits

191 Negative Binomial Interpret results the same was as Poisson Look at coefficient/standard error on delta Ho: delta = 0 (Poisson model is correct) In this case, delta = 5.21 standard error is 0.15, easily reject null. Var/Mean = 1+delta = 6.21, Poisson is mis-specificed, should see very small standard errors in the wrong model

192 Selected Results, Count Models Parameter (Standard Error) VariablePoissonNegative Binomial Age (0.026)0.103(0.055) Age (0.026)0.204(0.054) Chronic0.500(0.014)0.509(0.029) Excel-0.784(0.031)-0.527(0.059) Ln(Inc).0.021(0.007)0.038(0.016)

193 Section Duration Data

194 Introduction Sometimes we have data on length of time of a particular event or ‘spells’ –Time until death –Time on unemployment –Time to complete a PhD Techniques we will discuss were originally used to examine lifespan of objects like light bulbs or machines. These models are often referred to as “time to failure”

195 Notation T is a random variable that indicates duration (time til death, find a new job, etc) t is the realization of that variable f(t) is a PDF that describes the process that determines the time to failure CDF is F(t) represents the probability an event will happen by time t

196 F(t) represents the probability that the event happens by ‘t’. What is the probability a person will die on or before the 65 th birthday?

197 Survivor function, what is the chance you live past (t) S(t) = 1 – F(t) If 10% of a cohort dies by their 65 th birthday, 90% will die sometime after their 65 th birthday

198 Hazard function, h(t) What is the probability the spell will end at time t, given that it has already lasted t What is the chance you find a new job in month 12 given that you’ve been unemployed for 12 months already

199 PDF, CDF (Failure function), survivor function and hazard function are all related λ(t) = f(t)/S(t) = f(t)/(1-F(t)) We focus on the ‘hazard’ rate because its relationship to time indicates ‘duration dependence’

200 Example: suppose the longer someone is out of work, the lower the chance they will exit unemployment – ‘damaged goods’ This is an example of duration dependence, the probability of exiting a state of the world is a function of the length

201 Mathematically d λ(t) /dt = 0 then there is no duration dep. d λ(t) /dt > 0 there is + duration dependence the probability the spell will end increases with time d λ(t) /dt < 0 there is – duration dependence the probability the spell will end decreases over time

202 Your choice, is to pick values for f(t) that have +, - or no duration dependence

203 Different Functional Forms Exponential –λ(t)= λ –Hazard is the same over time, a ‘memory less’ process Weibull –F(t) = 1 – exp(-γt α ) where α,γ > 0 –λ(t) = αγt α-1 –if α>1, increasing hazard –if α<1, decreasing hazard –if α=1, exponential

204 Others: Lognormal, log-logistic, Gompertz

205 NHIS Multiple Cause of Death NHIS –annual survey of 60K households –Data on individuals –Self-reported healthm DR visits, lost workdays, etc. MCOD –Linked NHIS respondents from to National Death Index through Dec 31, 1995 –Identified whether respondent died and of what cause

206 Our sample –Males, 50-70, who were married at the time of the survey – surveys –Give everyone 5 years (60 months) of followup

207 Key Variables max_mths maximum months in the survey. Diedin5 respondent died during the 5 years of followup Note if diedn5=0, the max_mths=60. Diedin5 identifies whether the data is censored or not.

208 Identifying Duration Data in STATA Need to identify which is the duration data stset length, failure(failvar) Length=duration variable Failvar=1 when durations end in failure, =0 for censored values If all data is uncensored, omit failure(failvar)

209 In our case Stset max_mths, failure(diedin5)

210 Getting Kaplan-Meier Curves Tabular presentation of results sts list Graphical presentation sts graph Results by subgroup sts graph, by(income)