The Assessment of Fit in the Class of Logistic Regression Models: A Pathway out of the Jungle of Pseudo-R²s

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The Assessment of Fit in the

Class of Logistic Regression

Models:

A Pathway out of the Jungle

of Pseudo-R²s

Dr. Wolfgang Langer

Martin-Luther-University of Halle-Wittenberg

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Contents:

G 1. What is the problem?

G 2. Summary of the econometric Monte-Carlo

studies for Pseudo-R2s

G 3. The generalization of the McKelvey-Zavoina

Pseudo-R2 for multinomial logits

G 4. An application in an election study of East

German pupils

G 5. What’s to do in future methodological

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1. What is the problem ?

Situation now:

G An increasing number of people uses logistic

models for qualitative dependent variables.

G There is no general agreement on how to assess

the fit corresponding to practical significance.

G Where is the pathway out of the jungle of the

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Amemiya (1981):

Recommendations

G 1. There is not any universial pseudo

coefficient of determination.

G 2. Most of the Pseudo-R²s have values outside

the range of [0;1]. They are not completely standardized.

G 3. Assessing the model fit you have to use 2

or 3 Pseudo-R²s basing on different construction criteria.

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What is the situation now ?

J.Scott Long: (1997):

- Regression Models for Categorical and Limited Dependent Variables. SAGE

- (2000): Scalar Measures of Fit for Regression Models. (http://www.indiana.edu/~jsl650/ )

1.He presents a lot of Pseudo-R2s discussed in

econometric literature.

2.He recommends the McKelvey-Zavoina

Pseudo-R2 as the best fit measure for binary and ordinal

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2.Summary of the econometric Monte-Carlo

studies for Pseudo-R

2

s

G

Econometricians have made a lot of

Monte-Carlo studies in the early 90s:

- Hagle & Mitchell 1992

- Veall & Zimmermann 1992, 1993, 1994

- Windmeijer 1995

G

They have systematically tested the most

common Pseudo-R² for binary and ordinal

probit / logit models

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How does a Monte-Carlo simulation study

work?

G 1. You have to generate 100 data sets with a

multinormal density and a prescribed multiple correlation. Then you have to

manipulate systematically the sample size and the multiple correlation.

G 2. You have to estimate the multiple linear

regression models for each cell of your design.

G 3. You have to dichotomize your dependent

variable using systematically different cutpoints.

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G 4. You have to estimate your binary probit or logit models.

G 5. You have to calculate your Pseudo-R²s for

each probit or logit model.

G 6. You have to compare the Pseudo-R²s of the

probit / logit model with the “true R²” of the linear regression model.

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Which Pseudo-R²s have been tested?

G Likelihood-based measures:

- Maddala / Cox-Snell Pseudo-R²

- Cragg-Uhler / Nagelkerke Pseudo-R²

G Log-Likelihood-based measures:

- McFadden Pseudo-R²

- Aldrich-Nelson Pseudo-R²

- Aldrich-Nelson Pseudo-R² with the Veall-Zimmermann correction

G Basing on the estimated probabilities:

- Efron / Lave Pseudo-R²

G Basing on the variance decomposition of the

estimated Probits / Logits:

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Maddala Pseudo R 2 1 L0 LA 2 n Theoretical range: 0 Maddala Pseudo R 2 1 ( LA ) 2 n Notation :

L0: Likelihood of the zero model

(with a regression constant only)

LA: Likelihood of the alternative model

n: Sample size

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Cragg Uhler Pseudo R 2 LA 2 n L 0 2 n 1 L0 2 n Theoretical range:

0 Cragg Uhler Pseudo R 2 1

Standardization of the Maddala Pseudo R 2

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McFadden Pseudo R 2 1 ln LA ln L0

Theoretical range:

0 McFadden Pseudo R 2 1

It does not reach its maximum of 1 !

Rule of Thumb: 0.20 McFadden Pseudo R 2 0.40

The model has an excellent fit ! (McFadden 1979, 307) Notation :

ln LA: Log Likelihood of the alternative model

ln L0: Log Likelihood of the zero model

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For Probits:

Aldrich Nelson Pseudo R2 LR

2 (MA) LR 2(M A) n(M A) 2(lnLA lnL0) 2(lnLA lnL0) n(M A) For Logits:

Aldrich Nelson Pseudo R2 LR

2 (MA) LR 2(M A) (n(MA)3.29) 2(lnLA lnL0) 2(lnLA lnL0) (n(M A)3.29 )

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T heo re tica l rang e :

Probits : 0 Aldrich N elson Pseu do R 2 2 lnL0 n 2 lnL0 Lo gits : 0 Aldrich N elson Pseu do R 2 2 lnL0

3 .29 n 2 lnL0 T he A N Pseu do R 2 de pen ds on the nu m ber an d the

prop ortio ns of the cate gories of the de pen dent variables! You can ea sily see this at the form u la of the log like liho od of the zero m o del:

ln L0 K k1 nk ln nk n N otation :

nk: N umbe r of cases of cate gory k of the de pen dent variable

n: Sam ple size

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Veall Zimmermann correction of the Aldrich Nelson Pseudo R2:

It is standardized by its own reachable maximum depending on k categories and their proportions range [0 ;1 ]

For Probits:

Aldrich Nelson Pseudo R2V&Z

2(lnLA lnL0) 2(lnLA lnL0) n

2lnL0 n 2lnL0 For Logits :

Aldrich Nelson Pseudo R2V&Z

2(lnLA lnL0)

2(lnLA lnL0) 3.29n

2lnL0

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La ve / Efron Ps eudo R 2 1 n i1 ( Yi pˆi)2 n i1 ( Yi Y )¯ 2 , w ith Y¯ 1 n n i1 Yi N otation:

Yi: Value of the de pe ndent v ariab le in c as e i

ˆ

p i: Es tim ate d pro bab ility for Y1 in c as e i ¯

Y: O bserved pro portio n of c ate go ry "1 " of the de pe ndent v ariab le

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McKelvey Zavoina Pseudo R2 SSRegression SSErrors SSRegression (For PROBITs) n i1 ( ˆy yˆ)2 (n1.0) n i1 ( ˆy yˆ)2

G Basing on the variance decomposition of the estimated Probits / Logits

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McKelvey Zavoina Pseudo R2 SSRegression SSErrors SSRegression (For LOGITs) n i1 ( ˆyi yˆ)2 (n3.29) n i1 ( ˆyi yˆ)2 N o t a t i o n : ˆ y i : E s t i m a t e d L o g i t / P r o b i t o f c a s e i ( t h e l i n e a r p r e d i c t o r ) 1 . 0 : 2 o f t h e n o r m a l P D F 3 . 2 9 : 2 o f t h e l o g i s t i c P D F Theoretical range: [ 0 ; 1]

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Results of the Monte-Carlo-Studies

Corresponding to binary and ordinal probits / logits:

G The McKelvey-Zavoina Pseudo-R² is the best

estimator for the “true R²s” of the OLS regression.

G The Aldrich-Nelson Pseudo-R² with the

Veall-Zimmermann correction is the best

approximation of the McKelvey-Zavoina Pseudo-R².

G Lave / Efron, Aldrich-Nelson, McFadden and

Cragg-Uhler Pseudo-R² severely underestimate the “true R²”.

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(1) ln P2i P1i ß021 K k1 ßk 21xki 21 (2) ln P3i P1i ß031 K k1 ßk 31xki 31

Assumptions for the error terms k1:

1. 21 and 31 are stochastically independent.

2. 21 and 31 are identically distributed.

(I I D Assumption : Hensher &Johnson 1981, 154)

Estimation equations of a multinomial logit model with 3 alternatives:

3. The generalization of the McKelvey&Zavoina Pseudo-R²s for multinomial logits

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What is really a multinomial logit (MNL)?

G It is a nonlinear multi-equation model.

G The error terms 21 and 31 are independent and

identically distributed.

G Therefore you can separately calculate the

McKelvey-Zavoina Pseudo-R² for each equation / comparison of alternatives !

G Resulting options:

- .1 Simultaneous estimation with the MNL - .2 Separate estimation of binary logistic

regression models for each comparison (Begg & Gray 1984)

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4.An application in an election study of East German students (Sachsen-Anhalt 1998)

Social Democratic Party (SPD)

Christian Democratic Union (CDU)

Party of Democratic Socialism (PDS, Post-Communist Party)

Free Democratic Party (FDP)

The Green Party / Alliance 90 (B90)

Right extremist parties (DVU, REP, NPD)

Dependent variable:

Which party do you prefer to vote for ? Alternatives: (one vote only)

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• AGE in years

• GENDER: boys vs. girls

• SCHOOL TYPE: Secondary school, GRAMMAR school, VOCATIONAL school

• Internal and external political EFFICACY

• Perceived influence of the peers on the vote (PEERS)

• Perceived influence of the parents on the vote (PARENTS)

• Perceived influence of the media on the vote (MEDIA)

• Perceived influence of the teachers on the vote (TEACHERS)

• Countryside vs. City (LOCATION)

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SPD 46,9% CDU 19,5% PDS 12,6% FDP 3,1% B90 6,9% DVU/REP/NPD 11,1% ( n = 1894 )

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Table 1: Party preference of pupils in Sachsen-Anhalt 1998. Multinomial Logit Model (LIMDEP 7 NT) Independent variables: Comparison: CDU vs. SPD Comparison: PDS vs. SPD Comparison: FDP vs. SPD Comparison: B90 vs. SPD Comparison: DVU,REP,NPD vs. SPD

Logit S.E. Logit S.E. Logit S.E. Logit S.E. Logit S.E.

Constant - 0,577 0,605 - 3,922** 0,755 - 6,387** 1,436 +0,265 0,906 - 6,562** 0,871 AGE - 0,043 0,035 +0,118** 0,042 +0,178* 0,077 - 0,052 0,053 +0,206** 0,047 GENDER +0,511** 0,131 +0,383* 0,154 +0,520 0,285 - 0,0002 0,200 +1,275** 0,188 GRAMMER VOCATIONAL +0,870** +0,756** 0,210 0,296 +0,932** +0,166 0,254 0,356 +0,898 - 0,214 0,467 0,655 +1,082** - 0,388 0,292 0,546 - 0,628 - 0,327 0,346 0,371 EFFICACY - 0,011 0,021 +0,049 0,025 +0,087 0,048 - 0,084** 0,032 +0,109** 0,029 PEERS - 0,031 0,084 +0,024 0,100 +0,060 0,183 +0,062 0,131 +0,838** 0,097 PARENTS +0,026 0,070 +0,062 0,082 - 0,034 0,154 - 0,164 0,111 - 0,488** 0,102 MEDIA - 0,146* 0,066 - 0,117 0,079 - 0,247 0,143 - 0,300** 0,102 - 0,219* 0,086 TEACHERS - 0,072 0,084 - 0,302** 0,108 - 0,226 0,201 - 0,063 0,137 - 0,032 0,108 LOCATION +0,296 0,187 +0,359 0,236 +0,232 0,447 -0,616* 0,304 +0,699** 0,246

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Tab le 2: Co mp aris o n: C DU vs. SPD Co mp aris on : PD S vs . SPD Co mp aris on : FD P vs . SPD C o mp ariso n: B 9 0 v s. SP D Co mp aris on : D VU,R EP,N PD vs. SPD Ov erall M cKelvey& Zavo in a Pseudo R ² 5 ,64 2 % 10 ,0 48 % 1 5,0 96 % 1 8, 77 9 % 34 ,99 8 % In d ep en dent v ariab les û-M& Z Ps eu do R ² in % û-M & Z Pseu do R² in % û-M & Z Pseu do R ² in % û-M& Z Pseudo R ² in % û-M & Z Pseud o R² in % A GE 0,2 26 3 ,4 60 5,0 82 1, 47 3 2 ,91 3 GENDER 2,0 53 0 ,5 95 0,8 94 0, 05 2 8 ,70 9 S CH OOL TYP E 2,0 71 2 ,2 14 2,6 61 1 1, 08 4 8 ,10 5 EFF ICA C Y 0,0 41 0 ,6 71 2,2 05 1, 69 9 2 ,07 7 P EER S 0,0 78 0 ,0 41 0,0 03 0, 02 7 7 ,46 6 PA RTENTS 0,0 26 0 ,0 00 0,2 77 0, 62 4 3 ,85 0 M EDIA 0,7 01 0 ,7 06 1,8 44 2, 11 0 0 ,77 3 TEA C HER S 0,0 70 1 ,4 70 0,7 74 0, 04 8 0 ,09 3 LOCA TI ON 0,3 08 0 ,4 46 0,1 66 1, 50 9 0 ,68 0 M od el f it:

Likelih o od- Ratio ²-Test = 4 52 ,29 F. G.= 5 0 . = 0 ,00 n = 1 89 4 ln L M0 = - 27 82, 79 ln L MA = -25 56 ,64

M cFad den -Pseud o R ² = 8 ,1 27 %

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0 5 10 15 20 25 30 35 D e l t a -M c K e l v e y & Z a v o i n a P s e u d o -R 2

CDU PDS FDP B90 DVU / REP / NPD

Vote for party in comparison with alternative SPD

Party preferences of pupils in Sachsen-Anhalt 1998: Overall fits and relative effect sizes of the

independent variables LOCATION TEACHERS MEDIA PARENTS PEERS EFFICACY SCHOOLTYPE GENDER AGE

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Comparision the overall fit of the multinomial and binary logit models by their McKelvey-Zavoina Pseudo-R²s

y = 1,117x - 1,229 R2 = 0,995 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 Multinomial Logit B ina ry Lo g it

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Comparison of the partial McKelvey-Zavoina Pseudo-R²s of the

multinomial and the binary logit models y = 1,025x + 0,037 R2 = 0,979 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 11 12 Multinomial Logits Bi nar y Logi ts

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5.What’s to do in future methodological studies?

G Researchers have to present the LR-X², the

log-likelihoods of their alternative and zero binary or ordinal logit models and the sample size.

G ==> You can easily calculate the corrected

Aldrich-Nelson Pseudo-R² with these scalars. Then you can realistically assess the fit of their binary or ordinal logit / probit model.

G Comparing different models you have to adjust

McKelvey-Zavoina Pseudo-R²s for the sample size and the number of estimated parameters without the constant (under investigation).

G Bootstrap studies might be helpful to analyse the

distribution of the M-Z Pseudo-R² and to construct its confidence intervals (under investigation).

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References:

GAldrich, J.H.& Nelson, F.D. (1984):

Linear probability, logit, and probit models. Newbury Park: SAGE (Quantitative Applications in the Social Sciences, 45)

GAmemiya, T. (1981):

Qualitative response models: a survey. Journal of Economic Literature, 21, pp.1483-1536

GBegg, C.B. & Gray, R. (1984):

Calculation of polychotomous logistic regression parameters using individualized regression. Biometrika, 71, pp.11-18

GCragg, S.G.& Uhler, R. (1970):

The demand for automobiles. Canadian Journal of Economics, 3, pp. 386-406

GEfron, B. (1978):

Regression and Anova with zero-one data. Measures of residual variation. Journal of American Statistical Association, 73, pp. 113-121

GHagle, T.M. & Mitchell II,G.E. (1992):

Goodness of fit measures for probit and Logit. American Journal of Political

Science, 36, 3, pp. 762-784

GHensher, D.A.& Johnson, L.W. (1981):

Applied discrete choice modelling. London: Croom Helm

GMaddala, G.S. (1983):

Limited-dependent and qualitative variables in econometrics. Cambridge: Cambridge University Press

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GMcFadden, D. (1979):

Quantitative methods for analysing travel behaviour of individuals: some recent developments. In: Hensher, D.A.& Stopher, P.R.: (eds):Behavioural travel

modelling. London: Croom Helm, pp. 279-318

GMcKelvey, R. & Zavoina, W. (1975):

A statistical model for the analysis of ordinal level dependent variables. Journal of Mathematical Sociology, 4, pp. 103-20

GNagelkerke, N.J.D. (1991):

A note on a general definition of the coefficient of determination. Biometrika, 78, 3, pp.691-693

GRonning, G. (1991):

Mikro-Ökonometrie. Berlin: Springer

GVeall, M.R. & Zimmermann, K.F. (1992):

Pseudo-R2 in the ordinal probit model. Journal of Mathematical Sociology, 16, 4, pp. 333-342

G Veall, M.R. & Zimmermann, K.F. (1994):

Evaluating Pseudo-R2's for binary probit models. Quality&Quantity, 28, pp. 151- 164

GWindmeijer, F.A.G. (1995):

Goodness-of-fit measures in binary choice models. Econometric Reviews, 14, 1, pp. 101-116

GZimmermann, K.F. (1993):

Goodness of fit in qualitative choice models: review and evaluation. In: Schneeweiß, H. & Zimmermann, K. (eds):

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