Transcript Using Quasi-variance to Communicate Sociological Results
Using Quasi-variance to Communicate Sociological Results from Statistical Models
Vernon Gayle & Paul S. Lambert
University of Stirling
Gayle and Lambert (2007) Sociology, 41(6):1191-1208.
“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).
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
Table 1: Logistic regression prediction that self rated health is ‘good’
(Parameter estimates for model 1 )
1 2
Beta Standard Error
3
Prob.
No Higher qualifications Higher Qualifications Males Females North East North West Yorkshire & Humberside East Midlands West Midlands East of England South East South West Inner London Outer London Constant 0.65
-0.20
0.09
0.12
0.15
0.13
0.32
0.36
0.26
0.17
0.27
0.48
4
95% Confidence Intervals
0.0056
0.0041
-
0.0102
0.0107
0.0111
0.0106
0.0107
0.0101
0.0109
0.0122
0.0111
0.0090
<.001
<.001
<.001
<.001
<.001
<.001
<.001
<.001
<.001
<.001
<.001
<.001
0.64
-0.21
0.07
0.10
0.13
0.11
0.29
0.34
0.24
0.15
0.25
0.46
0.66
-0.20
0.11
0.14
0.17
0.15
0.34
0.38
0.28
0.20
0.29
0.50
Beta Standard Error Prob.
North East North West Yorkshire & Humberside 0.09
0.12
-
95% Confidence
-
0.07
0.10
Intervals
-
0.11
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 -
t
ˆ 2 s.e.
( ˆ 2 ˆ 3 ˆ 3 ) The banana skin is here!
Standard Error of the Difference
var( ˆ 2 ) var( ˆ 3 ) 2 (cov ( ˆ 2 ˆ 3 )) 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 Row 1
North West North West .00010483
Yorkshire & Humberside East Midlands West Midlands East England South East South West Inner London
2 3 4 5 6
Yorkshire & Humberside East Midlands West Midlands East England South East .00007543
.00011543
.00007543
.00007543
.00007543
.00007543
.00007544
.00007543
.00007545
.00007544
.00012312
.00007543
.00011337
.00007543
.00007544
.00007543
.00007544
Covariance
.0001148
.00007545
.00010268
9
Outer London
7
South West .00007544
.00007543
.00007544
.00007543
.00007544
.00007546
.00011802
8 9
Inner London Outer London .00007552
.00007547
.00007548
.00007545
.0000755
.00007547
.00007546
.00007545
.00007554
.00007572
.00007548
.00007555
.00007558
.00015002
.00007549
.00007598
.00012356
Standard Error of the Difference
0.0083 = 0.00010483
0.00011543
2 ( 0.00007543
) Variance North West (s.e.
2 ) Variance Yorkshire & Humberside (s.e.
2 ) Only Available in the variance covariance matrix
Formal Tests
t = -0.03 / 0.0083 = -3.6
Wald
c
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)
s.e. difference ≈
quasi
var( ˆ 2 )
quasi
var( ˆ 3 )
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
Beta
2
Standard Error
3
Prob.
4
95% Confidence Intervals
5
Quasi Variance
No Higher qualifications Higher Qualifications 0.65
0.0056
<.001
0.64
0.66
Males Females North East North West Yorkshire & Humberside -0.20
0.0041
<.001
-0.21
-0.20
0.09
0.12
0.0102
0.0107
<.001
<.001
0.07
0.10
0.11
0.14
0.0000755
0.0000294
0.0000400
Firth’s Method (made simple)
s.e. difference ≈
quasi
var( ˆ 2 )
quasi
var( ˆ 3 ) 0.0083 =
0.0000294
0.0000400
t = (0.09-0.12) / 0.0083 = -3.6 Wald
c
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
Row Column 1 2 3 4 5 6 7 8
North West Yorkshire & Humberside East Midlands West Midlands East England South East South West Inner London
9
Outer London
1
North West .00010483
2 3 4 5
Yorkshire & Humberside East Midlands West Midlands .00007543
.00007543
.00007543
East England .00007544
.00011543
.00007543
.00007543
.00007543
.00012312
.00007543
.00007543
.00011337
.00007543
.0001148
6
South East .00007545
.00007544
.00007544
.00007544
.00007545
.00010268
7
South West .00007544
.00007543
.00007544
.00007543
.00007544
.00007546
.00011802
8 9
Inner London Outer London .00007552
.00007548
.0000755
.00007547
.00007554
.00007572
.00007558
.00015002
.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
Conclusion – We should start using method
Benefits Costs • Overcomes the reference category problem when presenting models • Extra column in results • Provides reliable results (even though based on an approximation) • Time convincing colleagues that this is a good thing • Easy(ish) to calculate
Conclusion –
Why have we told you this… • Categorical X vars are ubiquitous • Interpretation of coefficients is critical to sociological analyses – Subtleties / slipperiness – (cf. in Economics where emphasis is often on precision rather than communication)