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Lovely Lucid Logistics
the analysis and graphic presentation of effects of
nominal and metric variables on binary outcomes
Diana Eugenie Kornbrot
Blended Learning Unit
University of Hertfordshire
[email protected]
11-Nov-05
Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
1
Abstract
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Logistic regression can be used to answer the same
questions about binary variables that ANOVA and
ANCOVA answer about metric variables.
However, SPSS provides much less support for
logistic regression. The Logistic Regression
Procedure provides no equivalent of ANOVA Means
Tables or Profile Plots.
This presentation shows how to use a combination of
SPSS Procedures to produce Tables and Graphs of
predicted logit and probabilities as a function of
categorical factor and metric covariate variables.
Diagnostics for model fit NOT discussed
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Merits own presentation
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 2
Acknowledgments
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Lia Kvavilashvili
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For all the prospective memory data
Stimulating theoretical discussion on content
ESRC Project Grant
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 3
Goals
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Motivate Logistic regression
Graphic Presentation of Logistic Model Results
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Predictions
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Factors and Contrasts
Application to Different Designs
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Logits and Probabilities as function explanatory variables
Identification of statistically reliable effects
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Interpretation much easier from graphs
Explanatory variables: 2 or 3 categorical
Explanatory variables: 1 metric, 1 or 2 categorical
Recommendations to Users of Logistic Regression
Recommendations to SPSS
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 4
Why Logistic Analysis?
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Need to analyse binary, i.e. 2 alternative,
responses
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Errors:
right, wrong
Events:
remembered, forgotten
Success: grant awarded, grant rejected
patient recovered, or not
More than 1 categorical variable
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Chi-square not sufficient
Combination of metric and categorical explanatory
variables
Interactions matter
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 5
Why Interpretation of Results is a Problem
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Analysis is on log (odds ratio) or logits
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Need for Packages SPSS or other
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Lack of intuitive feel for logits
Lack of intuitive feel for odds ratios for non-betters
Probabilities are more ‘natural’?
Can’t hand calculate, as no closed form answer
SPSS Output
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Primary output is in logits
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No directly useful graphics output
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BUT Save permits direct saving of probabilities no logits
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?No confidence levels on probabilities
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 6
Analysis
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Analysis
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GLM framework
Effects assumed to be linear on logits
Model Goodness of Fit Test on – 2LogLikelihood, -2LL
Model Fitting Procedure
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Effect of Evauluation Criteria: SPSS uses Wald
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SPSS uses Wald, other packages use deviance = -2LL
On factors and covariates
On model parameters
Other Packages Vary, all give Wald as minimum
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JMP, SPSS, SAS, SYSTAT
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 7
Data Example: Prospective Memory
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Prospective Memory
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Does person have GOOD prospective memory
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5 or 6 occasions remembered from 6 opportunities
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Model 1: task(action, event, time), age(4 categories)
Model 2: task(action, event, time), age(4), intellect
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Presentation Criteria
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Easy to interpret > Graphics
Predicted probability and logits
Estimate of accuracy as part of results
Tests for explanatory variable effects and contrasts
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 8
Model 1 using SPSS menus
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Analyze > Regression > Binary Logistic
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Dependent
Covariates
Method
Categorical
or
Save
Options
good#
task#(cat)
age#(cat)
task#(cat)*age#(cat)
Enter
task#(deviation) age#(deviation)
task#(repeated) age#(repeated)
!!!NOT indicator, the default!!!
not a lot of people know that!
probabilities, Cook’s, deviation
CI for exp(B)
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 9
Model 1 Global Results
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Model 1: task(action, event, time), age(4 categories)
Omnibus Test Significant = Good
Model Summary Substantial variance accounted for
Model Summary
Omnibus Tests of Mo del Co efficients
Chi-square
Step 1 Step
48.948
Block
48.948
Model
48.948
df
11
11
11
Sig.
.000
.000
.000
Step
1
-2 Log
Cox & Snell
likelihood
R Square
180.197 a
.238
Nagelkerke
R Square
.331
a. Estimation terminated at iteration number 6 because
parameter estimates changed by less than .001.
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 10
SPSS: Model 1 Parameters
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Variable effect not salient
No effects or standard errors for reference (last)
Wald Estimates of s.e. may not be those that are needed?
Variables in the Equation
B
Step
a
1
TASK#
TASK#(1)
TASK#(2)
AGE#
AGE#(1)
AGE#(2)
AGE#(3)
AGE# * TASK#
AGE#(1) by TASK#(1)
AGE#(1) by TASK#(2)
AGE#(2) by TASK#(1)
AGE#(2) by TASK#(2)
AGE#(3) by TASK#(1)
AGE#(3) by TASK#(2)
Constant
S.E.
.828
-1.024
.299
.259
1.309
.064
-.303
.371
.335
.344
.290
-.478
.014
.257
-.536
-.476
.704
.600
.441
.512
.442
.480
.469
.199
Wald
16.521
7.677
15.611
17.675
12.476
.036
.774
7.540
.234
1.175
.001
.337
1.246
1.028
12.554
df
2
1
1
3
1
1
1
6
1
1
1
1
1
1
1
Sig.
.000
.006
.000
.001
.000
.850
.379
.274
.629
.278
.978
.561
.264
.311
.000
Exp(B)
95.0% C.I.for EXP(B)
Lower
Upper
2.288
.359
1.274
.216
4.110
.597
3.704
1.066
.739
1.791
.553
.376
7.660
2.054
1.450
1.337
.620
1.014
1.293
.585
.621
2.022
.412
.261
.372
.543
.228
.248
4.337
1.472
2.766
3.075
1.500
1.559
a. Variable(s) entered on step 1: TASK#, AGE#, AGE# * TASK# .
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 11
SPSS Graphic Representation
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Predicted Probabilities, pre_1
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Directly Available from Save
Logits can be calculated
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Compute > Transform
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Graph > Interactive > Line plot
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Lgt = ln(pre_1/(1-pre_1)
NB Most other packages allow direct saving of logits
Y axis
X axis
Colour
predicted probability (mean)
age#
task#
No interactions
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So expect logit plots to be ‘more’ linear
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 12
SPSS: Logit & Probability Graphs
1.00
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0.80
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action
event
time
task#
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2.00
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action
event
time
0.60
lgt
Predicted probability
3.00
task#
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1.00
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0.40
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0.00
0.20
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-1.00
18-30
61-65
71-75
76-80
age#
18-30
61-65
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71-75
76-80
age#
Raw probability
Logit ??looks more linear??
Confidence Levels???
NOT in SPSS!!!
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 13
Confidence Levels
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Assume no extra-binomial dispersion
Asymptotic for logit
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Asymptotic for probability
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Symmetric about mean(lgt)
se(lgt)2 = 1/Noccur - 1/Nnot occur
Lower Confidence Level, 95%, LCL(lgt) = mean(lgt) -1.96se(lgt)
Upper Confidence Level, 95%, LCL(lgt) = mean(lgt) +1.96se(lgt)
Asymmetric about mean(prob).
Calculate from lgt CLs
probability = exp(lgt)/[1+exp(lgt]
LCL(prob) = exp(LCL(lgt)0/[1+exp(LCL(lgt))
UCL(prob) = exp(UCL(lgt)0/[1+exp(UCL(lgt))
Use EXCEL, can’t customise error bars in SPSS
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 14
EXCEL: Logit & Probability Graphs
action
event
time
1.0000
action
event
time
6.00
.9000
5.00
.8000
4.00
.7000
.6000
3.00
.5000
2.00
.4000
1.00
.3000
.00
.2000
18-30
61-65
71-75
76-80
-1.00
.1000
-2.00
.0000
18-30
61-65
Raw probability
71-75
76-80
-3.00
Logit
Errors are for each group. So low power for interaction
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 15
Model 2 Using SPSS menus
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Analyze > Regression > Binary Logistic
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Dependent
Covariates
Method
Categorical
or
Save
Options
good#
task#(cat), age#(cat), intellec
task#(cat)*age#(cat)
task#(cat)*intellec
intellec*age#(cat)
task#(cat)*age#(cat)*intellec
Enter
task#(deviation), age#(deviation)
task#(repeated), age#(repeated
probabilities, Cook’s, deviation
CI for exp(B)
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 16
Model 2 Summary
Omnibus=Whole Model LR chi2(23)=82.2, p=.0000001
Various r2 values
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McFadden=.36; Cox & Snell=.37; Nagelkerke=.51
Variable Effects
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Source
TASK
AGE 3
intellect
TASK*AGE
TASK*intellect
AGE*intellect
TASK*AGE*intellect
DF Wald chi^2
2
14.03
3
4.45
1
2.87
6
6.00
2
4.32
3
5.00
6
10.52
Wald Prob
.000899
.217040
.089995
.423621
.115183
.171542
.104480
LR Chi^2
29.70
4.96
6.03
14.63
7.73
7.07
21.43
LR Prob
.000000
.174500
.014101
.023371
.021003
.069614
.001532
Comparison of Variable Effects with different methods/packages
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1.
Likelihood Ratio shows strong effects intellec + intellec interactions
Used JMP-IN [even version 3, 5 is better for some things]
2.
3.
Wald does NOT show these effect - WORRYING
Model improvement with intellec: chi2(12)=33.3, p=.00087
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 17
Model 2 Probability by Age
18-30
Predicted probability: Model 2
1.00
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
61-65
  
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•Not very clear!
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0.75
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action
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event
time
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task#
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0.50
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0.25
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0.00
71-75
  
1.00
76-80
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Predicted probability: Model 2
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0.75
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0.50
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•Task effect:
•Event has lower prob
•Intellect:
•Most groups:
•Prob increase with intellec
•3 way interactions:
• > 70, event; 61-65 time
•Prob decrease with intellec
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0.25
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-2.00
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
0.00
-3.00

 
-1.00
0.00
intellec
1.00
2.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
intellec
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 18
Model 2 Logit by Age
18-30
61-65
25
20
action
event
time
15
10
lgt2
•Bit clearer!
task#
5
0
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    
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-5
-10
-15
71-75
76-80
25
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20
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15
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lgt2
10
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5
0
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-5
-10
-15
•Task effect:
•Event has lower prob
•Intellect:
•Most groups:
•Prob increase with intellec
•Large: 71-75time, 76-80action
•3 way interactions:
• > 70, event; 61-65 time
•Prob decrease with intellec

-3
-2
-1
0
intellec
1
2
-3
-2
-1
0
1
2
intellec
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 19
Summary & Recommendations
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Recommend Logit analyses as a very important tool
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Recommend Graphic displays toimprove interpretability
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SPSS provides basic procedure
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Limitations of SPSS
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No direct predicted logit or probability Table or Graph Summary
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Poor model diagnostics and power procedures
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No direct group standard errors
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No Maximum Likelihood estimates for explanatory variables
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No mixed models
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Other general packages are also DIRE - in different ways
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Need simple tools for routine logistic applications
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Can SPSS User Groups do anything?
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 20
References
Agresti, A. (1990). Categorical data analyses. Chichester: Wiley.
Agresti, A. (1996). Introduction to categorical data analyses. Chichester: Wiley.
Agresti, A., & Finley, B. (1997). Statistical methods for the social sciences (3 ed.). Upper
Saddle River, NJ: Prentice Hall.
Agresti, A., & Hartzel, J. (2000). Tutorial in biostatistics: strategies for comparing
treatments on a binary response with mulit-centre data. Statistics in Medicine, 19,
1115-1139.
Everitt, B., & Dunn, G. (2001). Applied multivariate data analysis (2 ed.). London: Edward
Arnold.
Kornbrot, D. E. (2000, 17-20 july 2000). Counting on prospective memory: Advantages of
logistic and log linear models over ANOVA and correlations. Paper presented at the
1st International Prospective Memory Conference, Hatfield, Hertfordshire, U.K.
Kvavilashvili, L., Kornbrot , D. E., Mash , V., Cockburn, J., & Milne, A. (2000, 17-20 july
2000). Remembering event-, time- and activity-based tasks in young, young-old and
old-old people. Paper presented at the 1st International Prospective Memory
Conference, Hatfield, Hertfordshire, U.K.
Lindsey, J. K. (1999). Models for repeated measurements (2 ed.). Oxford: Oxford
University Press.
Sofroniou, N., & Hutcheson, G. D. (2002). Confidence Intervals for the Predictions of
Logistic Regression in the Presence and Absence of a Variance– Covariance Matrix.
Understanding Statistics, 1(1), 3–18.
Tabachnick, B. G., & Fidell, L. S. (1996). Using multivariate statistics (3 ed.). New York:
Harper Collins.
11-Nov-05 Lucid Logistics: Kornbrot, Blended Learning Unit, University of Hertfordshire
SPSS York 21