issues and solutions in fitting, evaluating, and ... · florian jaeger, victor kuperman sample data...

Issues andSolutions inRegressionModeling

Florian Jaeger,Victor Kuperman

Sample Data andSimple Models

Building aninterpretablemodel

Model Evaluation

Reporting themodel

Discussion

Issues and Solutions in Fitting,Evaluating, and Interpreting Regression

Models

Florian Jaeger, Victor Kuperman

March 24, 2009





Model Evaluation

Reporting themodel

Discussion

Hypothesis testing in psycholinguisticresearch

I Typically, we make predictions not just about theexistence, but also the direction of effects.

I Sometimes, we’re also interested in effect shapes(non-linearities, etc.)

I Unlike in ANOVA, regression analyses reliably testhypotheses about effect direction and shape withoutrequiring post-hoc analyses if (a) the predictors in themodel are coded appropriately and (b) the model canbe trusted.

I Today: Provide an overview of (a) and (b).





Model Evaluation

Reporting themodel

Discussion

Overview

I Introduce sample data and simple modelsI Towards a model with interpretable coefficients:

I outlier removalI transformationI coding, centering, . . .I collinearity

I Model evaluation:I fitted vs. observed valuesI model validationI investigation of residualsI case influence, outliers

I Model comparisonI Reporting the model:

I comparing effect sizesI back-transformation of predictorsI visualization





Model Evaluation

Reporting themodel

Discussion

Sample Data and Simple Models

Building an interpretable modelData explorationTransformationCodingCenteringInteractions and modeling of non-linearitiesCollinearity

What is collinearity?Detecting collinearityDealing with collinearity

Model EvaluationBeware overfitting

Detect overfitting: ValidationGoodness-of-fit

Aside: Model Comparison

Reporting the modelDescribing PredictorsWhat to reportBack-transforming coefficientsComparing effect sizesVisualizing effectsInterpreting and reporting interactions

Discussion





Model Evaluation

Reporting themodel

Discussion

Data 1: Lexical decision RTs

I Outcome: log lexical decision latency RTI Inputs:

I factors Subject (21 levels) and Word (79 levels),I factor NativeLanguage (English and Other)I continuous predictors Frequency (log word frequency),

and Trial (rank in the experimental list).

Subject RT Trial NativeLanguage Word Frequency1 A1 6.340359 23 English owl 4.8598122 A1 6.308098 27 English mole 4.6051703 A1 6.349139 29 English cherry 4.9972124 A1 6.186209 30 English pear 4.7273885 A1 6.025866 32 English dog 7.6676266 A1 6.180017 33 English blackberry 4.060443





Model Evaluation

Reporting themodel

Discussion

Linear model of RTs

> lin.lmer = lmer(RT ~ NativeLanguage ++ Frequency + Trial ++ (1 | Word) + (1 | Subject),+ data = lexdec)

Random effects:Groups Name Variance Std.Dev.Word (Intercept) 0.0029448 0.054266Subject (Intercept) 0.0184082 0.135677Residual 0.0297268 0.172415

Number of obs: 1659, groups: Word, 79; Subject, 21

Fixed effects:Estimate Std. Error t value

(Intercept) 6.548e+00 4.963e-02 131.94NativeLanguageOther 1.555e-01 6.043e-02 2.57Frequency -4.290e-02 5.829e-03 -7.36Trial -2.418e-04 9.122e-05 -2.65

I estimates for random effects of Subject and Word andfor the residual error of the model: standard deviationand variance.

I estimates for regression coefficients, standard errors →t-values

I Effect significant if ± 2*SE does not include zero (ift-value of ± 2).





Model Evaluation

Reporting themodel

Discussion

Linear model of RTs (cnt’d)

I t-value anti-conservative

→ MCMC-sampling of coefficients to obtain nonanti-conservative estimates

> pvals.fnc(lin.lmer, nsim = 10000)

$fixedEstimate MCMCmean HPD95lower HPD95upper pMCMC Pr(>|t|)

(Intercept) 6.5476 6.5482 6.4653 6.6325 0.0001 0.0000NativeLanguageOther 0.1555 0.1551 0.0580 0.2496 0.0012 0.0001Frequency -0.0429 -0.0429 -0.0542 -0.0323 0.0001 0.0000Trial -0.0002 -0.0002 -0.0004 -0.0001 0.0068 0.0109

$randomGroups Name Std.Dev. MCMCmedian MCMCmean HPD95lower HPD95upper

1 Word (Intercept) 0.0564 0.0495 0.0497 0.0384 0.06192 Subject (Intercept) 0.1410 0.1070 0.1083 0.0832 0.13793 Residual 0.1792 0.1737 0.1737 0.1678 0.1799





Model Evaluation

Reporting themodel

Discussion

Data 2: Lexical decision responseI Outcome: Correct or incorrect response (Correct)I Inputs: same as in linear model

> lmer(Correct == "correct" ~ NativeLanguage ++ Frequency + Trial ++ (1 | Subject) + (1 | Word),+ data = lexdec, family = "binomial")

Random effects:Groups Name Variance Std.Dev.Word (Intercept) 1.01820 1.00906Subject (Intercept) 0.63976 0.79985Number of obs: 1659, groups: Word, 79; Subject, 21

Fixed effects:Estimate Std. Error z value Pr(>|z|)

(Intercept) -1.746e+00 8.206e-01 -2.128 0.033344 *NativeLanguageOther -5.726e-01 4.639e-01 1.234 0.217104Frequency 5.600e-01 1.570e-01 -3.567 0.000361 ***Trial 4.443e-06 2.965e-03 0.001 0.998804

I estimates for random effects of Subject and Word (noresiduals).

I estimates for regression coefficients, standard errors →Z- and p-values





Model Evaluation

Reporting themodel

Discussion

Interpretation of coefficients

I In theory, directionality and shape of effects can betested and immediately interpreted.

I e.g. logit model

Fixed effects:Estimate Std. Error z value Pr(>|z|)

(Intercept) -1.746e+00 8.206e-01 -2.128 0.033344 *NativeLanguageOther 5.726e-01 4.639e-01 1.234 0.217104Frequency -5.600e-01 1.570e-01 -3.567 0.000361 ***Trial -5.725e-06 2.965e-03 -0.002 0.998460

I . . . but can these coefficient estimates be trusted?





Data exploration

Transformation

Coding

Centering

Interactions and modelingof non-linearities

Collinearity

What is collinearity?

Detecting collinearity

Dealing with collinearity

Model Evaluation

Reporting themodel

Discussion

Sample Data and Simple Models

Building an interpretable modelData explorationTransformationCodingCenteringInteractions and modeling of non-linearitiesCollinearity

What is collinearity?Detecting collinearityDealing with collinearity

Model EvaluationBeware overfitting

Detect overfitting: ValidationGoodness-of-fit

Aside: Model Comparison

Reporting the modelDescribing PredictorsWhat to reportBack-transforming coefficientsComparing effect sizesVisualizing effectsInterpreting and reporting interactions

Discussion





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Modeling schema





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Data exploration





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Data exploration

I Select and understand input variables and outcomebased on a-priori theoretical consideration

I How many parameters does your data afford(yoverfitting)?

I Data exploration: Before fitting the model, exploreinputs and outputs

I Outliers due to missing data or measurement error (e.g.RTs in SPR < 80msecs).

I NB: postpone distribution-based outlier exclusion untilafter transformations)

I Skewness in distribution can affect the accuracy ofmodel’s estimates (ytransformations).





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Understanding variance associated withpotential random effects

I explore candidate predictors (e.g., Subject or Word) forlevel-specific variation.

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→ Huge variance.





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Random effects (cnt’d)

I explore variation of level-specific slopes.

Trial

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→ not too much variance.I random effect inclusion test via ymodel comparison





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Understanding input variablesI Explore:

I correlations between predictors (ycollinearity).I non-linearities may become obvious (lowess).

RT

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r = −0.06

p = 0.015

rs = −0.05

p = 0.037

r = 0

p = 0.9076

rs = 0

p = 0.8396

Trial

5010

015

0

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

6.0 6.5 7.0 7.5

1.0

1.4

1.8 r = 0.32

p = 0

rs = 0.32

p = 0

r = 0

p = 1

rs = 0

p = 1

50 100 150

r = −0.01

p = 0.5929

rs = −0.01

p = 0.5966

NativeLanguage

> pairscor.fnc(lexdec[,c("RT", "Frequency", "Trial", "NativeLanguage")])





Data exploration

Transformation

Coding

Centering


Collinearity




Model Evaluation

Reporting themodel

Discussion

Non-linearities

I Consider Frequency (already log-transformed inlexdec) as predictor of RT:

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issues and solutions in fitting, evaluating, and ... · florian jaeger, victor kuperman sample data...

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