Presenting results from statistical models

Professor Vernon Gayle and Dr Paul Lambert(Stirling University)

Wednesday 1st April 2009

Structure of the Seminar

Should take 1 semester!!!

1. Principals of model construction and interpretation

2. Key variables – measurement and func. Form

3. Presenting results

4. Longitudinal data analysis

5. Individuals in households – multilevel models

“One of the useful things about mathematical and statistical models [of educational realities] is that, so long as one states the assumptions clearly and follows the rules correctly, one can obtain conclusions which are, in their own terms, beyond reproach. The awkward thing about these models is the snares they set for the casual user; the person who needs the conclusions, and perhaps also supplies the data, but is untrained in questioning the assumptions….

…What makes things more difficult is that, in trying to communicate with the casual user, the modeller is obliged to speak his or her language – to use familiar terms in an attempt to capture the essence of the model. It is hardly surprising that such an enterprise is fraught with difficulties, even when the attempt is genuinely one of honest communication rather than compliance with custom or even subtle indoctrination” (Goldstein 1993, p. 141).

Structure of the this session

1. Presenting results

• This talk could also take weeks on end

• Two topics only - not the final word

– Quasi-Variances– Sample Enumeration methods

• Many more topics emerging, – propensity score matching– simulation modelling

Using Quasi-variance to Communicate Sociological Results from Statistical Models

Vernon Gayle & Paul S. LambertUniversity of Stirling

Gayle and Lambert (2007) Sociology, 41(6):1191-1208

A little biography (or narrative)…

• Since being at Centre for Applied Stats in 1998/9 I has been thinking about the issue of model presentation

• Done some work on Sample Enumeration Methods with Richard Davies

• Summer 2004 (with David Steele’s help) began to think about “quasi-variance”

• Summer 2006 began writing a paper with Paul Lambert

The Reference Category Problem

• In standard statistical models the effects of a categorical explanatory variable are assessed by comparison to one category (or level) that is set as a benchmark against which all other categories are compared

• The benchmark category is usually referred to as the ‘reference’ or ‘base’ category

The Reference Category Problem

An example of Some English Government Office Regions

0 = North East of England

----------------------------------------------------------------

1 = North West England

2 = Yorkshire & Humberside

3 = East Midlands

4 = West Midlands

5 = East of England

Government Office Region

1 2 3 4

Beta StandardError

Prob. 95% Confidence Intervals

No Higher qualifications - - - - -

Higher Qualifications 0.65 0.0056 <.001 0.64 0.66

Males - - - - -

Females -0.20 0.0041 <.001 -0.21 -0.20

North East - - - - -

North West 0.09 0.0102 <.001 0.07 0.11

Yorkshire & Humberside 0.12 0.0107 <.001 0.10 0.14

East Midlands 0.15 0.0111 <.001 0.13 0.17

West Midlands 0.13 0.0106 <.001 0.11 0.15

East of England 0.32 0.0107 <.001 0.29 0.34

South East 0.36 0.0101 <.001 0.34 0.38

South West 0.26 0.0109 <.001 0.24 0.28

Inner London 0.17 0.0122 <.001 0.15 0.20

Outer London 0.27 0.0111 <.001 0.25 0.29

Constant 0.48 0.0090 <.001 0.46 0.50

Table 1: Logistic regression prediction that self-rated health is ‘good’ (Parameter estimates for model 1 )

Beta StandardError

Prob. 95% Confidence Intervals

North East - - - - -

North West 0.09 0.07 0.11

Yorkshire & Humberside 0.12 0.10 0.14

Conventional Confidence Intervals

• Since these confidence intervals overlap we might be beguiled into concluding that the two regions are not significantly different to each other

• However, this conclusion represents a common misinterpretation of regression estimates for categorical explanatory variables

• These confidence intervals are not estimates of the difference between the North West and Yorkshire and Humberside, but instead they indicate the difference between each category and the reference category (i.e. the North East)

• Critically, there is no confidence interval for the reference category because it is forced to equal zero

Formally Testing the Difference Between Parameters -

)ˆˆ( s.e.

ˆˆ

3-2

3-2

t

The banana skin is here!

Standard Error of the Difference

))ˆˆ( (cov 2 - )ˆvar( )ˆvar( 3-232

Variance North West (s.e.2 )

Variance Yorkshire & Humberside (s.e.2 )

Only Available in the variance covariance matrix

Table 2: Variance Covariance Matrix of Parameter Estimates for the Govt Office Region variable in Model 1

Column 1 2 3 4 5 6 7 8 9

Row North West

Yorkshire &Humberside

East Midlands

West Midlands

East England

South East South West Inner London

Outer London

1 North West .00010483

2 Yorkshire &Humberside

.00007543 .00011543

3 East Midlands

.00007543 .00007543 .00012312

4 West Midlands

.00007543 .00007543 .00007543 .00011337

5 East England

.00007544 .00007543 .00007543 .00007543 .0001148

6 South East .00007545 .00007544 .00007544 .00007544 .00007545 .00010268

7 South West .00007544 .00007543 .00007544 .00007543 .00007544 .00007546 .00011802

8 Inner London

.00007552 .00007548 .0000755 .00007547 .00007554 .00007572 .00007558 .00015002

9 Outer London

.00007547 .00007545 .00007546 .00007545 .00007548 .00007555 .00007549 .00007598 .00012356

Covariance

Standard Error of the Difference

Variance North West (s.e.2 )

Variance Yorkshire & Humberside (s.e.2 )

Only Available in the variance covariance matrix

)0.00007543 ( 2- 0.00011543 0.000104830.0083 =

Formal Tests

t = -0.03 / 0.0083 = -3.6

Wald 2 = (-0.03 /0.0083)2 = 12.97; p =0.0003

Remember – earlier because the two sets of confidence intervals overlapped we could wrongly conclude that the two regions were not significantly different to each other

Comment

• Only the primary analyst who has the opportunity to make formal comparisons

• Reporting the matrix is seldom, if ever, feasible in paper-based publications

• In a model with q parameters there would, in general, be ½q (q-1) covariances to report

Firth’s Method (made simple)

)ˆvar( )ˆvar( 32 quasiquasi s.e. difference ≈

Table 1: Logistic regression prediction that self-rated health is ‘good’ (Parameter estimates for model 1, featuring conventional regression results, and quasi-variance statistics )

1 2 3 4 5

Beta StandardError

Prob. 95% Confidence Intervals

Quasi-Variance

No Higher qualifications - - - - - -

Higher Qualifications 0.65 0.0056 <.001 0.64 0.66 -

Males - - - - - -

Females -0.20 0.0041 <.001 -0.21 -0.20 -

North East - - - - - 0.0000755

North West 0.09 0.0102 <.001 0.07 0.11 0.0000294

Yorkshire & Humberside 0.12 0.0107 <.001 0.10 0.14 0.0000400

Firth’s Method (made simple)

)ˆvar( )ˆvar( 32 quasiquasi s.e. difference ≈

0.0000400 0.0000294 0.0083 =

t = (0.09-0.12) / 0.0083 = -3.6

Wald 2 = (-.03 / 0.0083)2 = 12.97; p =0.0003

These results are identical to the results calculated by the conventional method

The QV based ‘comparison intervals’ no longer overlap

Firth QV Calculator (on-line)

Table 2: Variance Covariance Matrix of Parameter Estimates for the Govt Office Region variable in Model 1

Column 1 2 3 4 5 6 7 8 9

Row North West Yorkshire &Humberside

East Midlands

West Midlands

East England

South East South West Inner London

Outer London

1 North West .00010483

2 Yorkshire &Humberside

.00007543 .00011543

3 East Midlands

.00007543 .00007543 .00012312

4 West Midlands

.00007543 .00007543 .00007543 .00011337

5 East England .00007544 .00007543 .00007543 .00007543 .0001148

6 South East .00007545 .00007544 .00007544 .00007544 .00007545 .00010268

7 South West .00007544 .00007543 .00007544 .00007543 .00007544 .00007546 .00011802

8 Inner London

.00007552 .00007548 .0000755 .00007547 .00007554 .00007572 .00007558 .00015002

9 Outer London

.00007547 .00007545 .00007546 .00007545 .00007548 .00007555 .00007549 .00007598 .00012356

Information from the Variance-Covariance Matrix Entered into the Data Window (Model 1)

0

0 0.00010483

0 0.00007543 0.00011543

0 0.00007543 0.00007543 0.00012312

0 0.00007543 0.00007543 0.00007543 0.00011337

0 0.00007544 0.00007543 0.00007543 0.00007543 0.00011480

0 0.00007545 0.00007544 0.00007544 0.00007544 0.00007545 0.00010268

0 0.00007544 0.00007543 0.00007544 0.00007543 0.00007544 0.00007546 0.00011802

0 0.00007552 0.00007548 0.00007550 0.00007547 0.00007554 0.00007572 0.00007558 0.00015002

0 0.00007547 0.00007545 0.00007546 0.00007545 0.00007548 0.00007555 0.00007549 0.00007598 0.00012356

-10

12

3P

ara

met

er e

stim

ate

BlackChinese

IndianWhite

BangladeshiPakistani

Ethnicity

Parameter estimate 95% confidence intervalParameter estimate 95% QV compariosn intervals

Source: YCS Cohort 9, n=12789.Model: Logistic regression estimating '5+ GCSE Passes A*-C'.

5+ GCSE Passes Year 11Ethnicity Effects

QV Conclusion – We should start using method

Benefits

• Overcomes the reference category problem when presenting models

• Provides reliable results (even though based on an approximation)

• Easy(ish) to calculate

• Has extensions to other models

Costs

• Extra column in results

• Time convincing colleagues that this is a good thing

Example

Drew, D., Gray, J. and Sime, N. (1992)

Against the odds: The Education and Labour Market Experiences of Black Young People

Comparison of Odds

Greater than 1 “higher odds”

Less than 1 “lower odds”

Naïve Odds• In this model (after controlling for other

factors)

White pupils have an odds of 1.0

Afro Caribbean pupils have an odds of 3.2

• Reporting this in isolation is a naïve presentation of the effect because it ignores other factors in the model

A Comparison

Pupil with

4+ higher passes

White

Professional parents

Male

Graduate parents

Two parent family

Pupil with

0 higher passes

Afro-Caribbean

Manual parents

Male

Non-Graduate parents

One parent family

Odds are multiplicative

4+ Higher Grades 1.0 1.0Ethnic Origin 1.0 3.2Social Class 1.0 0.5Gender 1.0 1.0Parental Education 1.0 0.6No. of Parents 1.0 0.9

Odds 1.0 0.86

Naïve Odds

• Drew, D., Gray, J. and Sime, N. (1992) warn of this danger….

• …Naïvely presenting isolated odds ratios is still widespread (e.g. Connolly 2006 Brit. Ed. Res. Journal 32(1),pp.3-21)

• We should avoid reporting isolated odds ratios where possible!

Logit scale

• Generally, people find it hard to directly interpret results on the logit scale – i.e.

Log Odds, Odds, Probability

• Log odds converted to odds = exp(log odds)

• Probability = odds/(1+odds)

• Odds = probability / (1-probability)

Log Odds, Odds, ProbabilityOdds ln odds p

99.00 4.60 0.99

19.00 2.94 0.95

9.00 2.20 0.9

4.00 1.39 0.8

2.33 0.85 0.7

1.50 0.41 0.6

1.00 0.00 0.5

0.67 -0.41 0.4

0.43 -0.85 0.3

0.25 -1.39 0.2

0.11 -2.20 0.1

0.05 -2.94 0.05

0.01 -4.60 0.01

Odds are asymmetric – beware!

Divide by 4 rule

• Gelman and Hill (2008) suggest dividing coefficients from logit models by 4 as a guide for assessing the effects of the estimated for a given explanatory variable as a probability

• They assert that /4 provides a ‘rule of convenience’ for estimating the upper bound of the predictive difference corresponding to a unit change in the explanatory variable.

• Gelman and Hill (2008) are careful to report that this is an approximation and that it performs best near the midpoint of the logistic curve

• We believe that this has some merit as a rough and ready method of interpreting the effects of estimates and is a useful tool especially when tables of coefficients are rapidly flashed up at a conference presentation

Gelman, A. and J. Hill (2008) Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge: Cambridge University Press

Communicating Results (to non-technically informed audiences)

• Davies (1992) Sample Enumeration

• Payne (1998) Labour Party campaign data

• Gayle et al. (2002)

• War against the uninformed use of odds (e.g. on breakfast t.v.)

Sample Enumeration Methods

In a nutshell…

“What if” – what if the gender effect was removed

1. Fit a model (e.g. logit)

2. Focus on a comparison (e.g. boys and girls)

3. Use the fitted model to estimate a fitted value for each individual in the comparison group

4. Sum these fitted values and construct a sample enumerated % for the group

Naïve Odds

• Naïvely presenting odds ratios is widespread (e.g. Connolly 2006)

• In this model naïvely (after controlling for other factors)

Girls have an odds of 1.0Boys have an odds of .58

We should avoid this where possible!

Logit Model

• Example from YCS 11

(these pupils took GCSE in 2001)

y=1 5+ GCSE passes (A* - C)

X vars

gender; family social class (NS-SEC);

ethnicity; housing tenure; parental education; parental employment;

school type; family type

Naïve Odds

• Example from YCS 11(these pupils took GCSE in 2001)

• In this model naïvely (after controlling for other factors)

Girls have an odds of 1.0Boys have an odds of .66

We should avoid this where possible!

Sample Enumeration Results

Percentage with 5+ GCSE (A*-C)

All 52%

Girls 58%

Boys 47%

(Sample enumeration est. boys) (50%)

Observed difference 11%

Difference due ‘directly’ to gender 3%

Difference due to other things 8%

Pseudo Confidence Interval

Sample Enumeration

Male Effect

Upper Bound 50.32%

Estimate 49.81%

Lower Bound 49.30%

Bootstrapping to construct a pseudo confidence interval (1000 Replications)

Reference

• A technical explanation of the issue is given in

Davies, R.B. (1992) ‘Sample Enumeration Methods for Model Interpretation’ in P.G.M. van der Heijden, W. Jansen, B. Francis and G.U.H. Seeber (eds) Statistical Modelling, Elsevier

We have recently written a working paper on logit models

http://www.dames.org.uk/publications.html

Conclusion –Why have we told you this…

• Categorical X vars are ubiquitous

• Interpretation of coefficients is critical to sociological analyses– Subtleties / slipperiness– (e.g. in Economics where emphasis is often on

precision rather than communication)