Summer School

1
Summer School

• Week 2

2
Contents

• Logistic regression refresher
• Some familiar some less familiar polytomous
  models
• 1PL/2PL in Stata and R
• PCM/RSM/GRM in R
• Link IRT to CFA/UV in Mplus
• DIF/MIMIC in Mplus

3
Types of outcome

• Two categories
• Binary / dichotomous
• Ordered
• e.g. low birthweight (lt 2500g), height gt 6ft, age
  gt 70
• Unordered
• e.g. gender, car-ownership, disease status
• Presence of ordering is unimportant for binaries

4
Types of outcome

• 3 categories
• Polytomous
• Ordered (ordinal)
• Age (lt30,30-40,41)
• Likert items (str disagree, disagree, , str
  agree)
• Unordered (nominal)
• Ethnicity (white/black/asian/other)
• Pet ownership (none/cat/dog/budgie/goat)

5
Modelling options (LogR/IRT)
6
Binary Logistic Regression
7
Binary Logistic Regression
Probability of a positive response /
outcome given a covariate
Intercept
Regression coefficient
8
Binary Logistic Regression
Probability of a negative response
9
Logit link function

• Probabilities only in range 0,1
• Logit transformation is cts in range (inf,inf)
• Logit is linear in covariates

10
Simple example cts predictor

• Relationship between birthweight and head
  circumference (at birth)
• Exposure
• birthweight (standardized)

variable mean sd ----------------------
------ bwt 3381.5g 580.8g --------------
---------------
11
Simple example cts predictor

• Outcome
• Head-circumference 53cm

headcirc Freq.
--------------------------- 0
8,898 84.4 1 1,651
15.7 --------------------------- Total
10,549
12
Simple example cts predictor
The raw data doesnt show much
13
Simple example cts predictor
Logistic regression models the probabilities
(here shown for deciles of bwt)

• bwt_z_grp
• headcirc 0 1 2
  3 4
• -------------------------------------------------
  -------------------
• 0 1,006 993 1,050
  946 1,024
• 99.80 98.12 97.95
  96.04 93.35
• -------------------------------------------------
  -------------------
• 1 2 19 22
  39 73
• 0.20 1.88 2.05
  3.96 6.65
• -------------------------------------------------
  -------------------
• headcirc 5 6 7
  8 9 Total
• -------------------------------------------------
  ---------------------------
• 0 931 922 856
  688 381 8,797
• 89.95 84.98 81.84
  66.67 35.94 84.33
• -------------------------------------------------
  ---------------------------
• 1 104 163 190
  344 679 1,635

14
Simple example cts predictor
Increasing, non-linear relationship
15
Simple example cts predictor

• Logistic regression
  Number of obs 10432
• 
  LR chi2(1) 2577.30
• 
  Prob gt chi2 0.0000
• Log likelihood -3240.9881
  Pseudo R2 0.2845
• --------------------------------------------------
  ----------------------------
• headcirc Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• bwt_z 7.431853 .378579 39.38
  0.000 6.72569 8.212159
• --------------------------------------------------
  ----------------------------
• Or in less familiar log-odds format
• --------------------------------------------------
  ----------------------------
• headcirc Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• bwt_z 2.005775 .0509401 39.38
  0.000 1.905935 2.105616
• _cons -2.592993 .0474003 -54.70
  0.000 -2.685896 -2.50009

16
Simple example cts predictor
Fitted model logit scale
17
Simple example cts predictor
Fitted model logit scale
Cons -2.59
Slope 2.00
18
Simple example cts predictor
But alsoa logit of zero represents point at
which both levels of outcome are equally likely
19
Simple example cts predictor
Fitted model probability scale
20
Simple example cts predictor
Fitted model probability scale
Point at which curve changes direction
21
Simple example cts predictor
Observed and fitted values (within deciles of bwt)
22
LogR cts predictor - summary

• Logit is linearly related to covariate
• Gradient gives strength of association
• Intercept is related to prevalence of outcome
• 
  seldom used

• Non-linear (S-shaped) relationship between
• probabilities and covariate
• Steepness of linear-section infers
• strength of association
• Point at which curve changes direction is where
• P(u1X) P(u0X) can be thought of as
  the
• location is related to prevalence of
  outcome

23
LogR binary predictor

• Define binary predictor bwt 8lb
• 32 of the sample had a birthweight of 8lb
• Same outcome
• Head circumference gt 53cm
• Does being 8lb at birth increase the chance of
  you being born with a larger head?

24
Association can be cross-tabbed

• headcirc
• bwt_8lb 0 1 Total
• -------------------------------------------
• 0 6,704 384 7,088
• 94.58 5.42 100.00
• -------------------------------------------
• 1 2,093 1,251 3,344
• 62.59 37.41 100.00
• -------------------------------------------
• Total 8,797 1,635 10,432
• 84.33 15.67 100.00

25
Association can be cross-tabbed

• headcirc
• bwt_8lb 0 1 Total
• -------------------------------------------
• 0 6,704 384 7,088
• 94.58 5.42 100.00
• -------------------------------------------
• 1 2,093 1,251 3,344
• 62.59 37.41 100.00
• -------------------------------------------
• Total 8,797 1,635 10,432
• 84.33 15.67 100.00

Familiar with (67041251)/(2093384) 10.43
odds-ratio
26
Association can be cross-tabbed

• headcirc
• bwt_8lb 0 1 Total
• -------------------------------------------
• 0 6,704 384 7,088
• 94.58 5.42 100.00
• -------------------------------------------
• 1 2,093 1,251 3,344
• 62.59 37.41 100.00
• -------------------------------------------
• Total 8,797 1,635 10,432
• 84.33 15.67 100.00

Familiar with (67041251)/(2093384) 10.43
odds-ratio
However ln(67041251)/(2093384) 2.345 log
odds-ratio
27
Association can be cross-tabbed

• headcirc
• bwt_8lb 0 1 Total
• -------------------------------------------
• 0 6,704 384 7,088
• 94.58 5.42 100.00
• -------------------------------------------
• 1 2,093 1,251 3,344
• 62.59 37.41 100.00
• -------------------------------------------
• Total 8,797 1,635 10,432
• 84.33 15.67 100.00

Familiar with (67041251)/(2093384) 10.43
odds-ratio
However ln(67041251)/(2093384) 2.345 log
odds-ratio
and ln(384)/(6704) ln(0.057) -2.86
intercept on logit scale
28
Logit output (from Stata)

• . logit headcirc bwt_8lb
• Logistic regression
  Number of obs 10432
• 
  LR chi2(1) 1651.89
• 
  Prob gt chi2 0.0000
• Log likelihood -3703.6925
  Pseudo R2 0.1823
• --------------------------------------------------
  ----------------------------
• headcirc Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• bwt_8lb 2.345162 .063486 36.94
  0.000 2.220732 2.469592
• _cons -2.859817 .0524722 -54.50
  0.000 -2.962661 -2.756974
• --------------------------------------------------
  ----------------------------

29
What lovely output figures!
There is still an assumed s-shape on probability
scale although the curve is not apparent
Linear relationship in logit space
30
What lovely output figures!
Intercept -2.86
Slope 2.35
There is still an assumed s-shape on probability
scale although the curve is not apparent
Linear relationship in logit space
31
LogR binary predictor - summary

• The same maths/assumptions underlie the models
  with a binary predictor
• Estimation is simpler can be done from crosstab
  rather than needing ML
• Regression estimates relate to linear
  relationship on logit scale

32
Multinomial Logistic Regression
33
Multinomial Logistic Regression

• Typically used for non-ordinal (nominal) outcomes
• Can be used for ordered data (some information is
  ignored)
• 3 outcome levels
• Adding another level adds another set of
  parameters so more than 4 or 5 levels can be
  unwieldy

34
Multinomial Logistic Regression
where c0 a0 0
Here the probabilities are obtained by a
divide-by-total procedure
35
Examples

• Outcome head-circumference
• 4 roughly equal groups (quartiles)
• Ordering will be ignored
• headcirc4 Freq. Percent
• ---------------------------------------
• lt 49cm 2,574 24.4
• 49.150.7cm 2,655 25.2
• 50.851.9cm 2,260 21.4
• 52 cm 3,060 29.0
• ---------------------------------------
• Total 10,549 100.00
• Exposure 1 birthweight of 8lb or more
• Exposure 2 standardized birthweight

36
Exposure 1 bwt gt 8lb

• 32 of the sample had a birthweight of 8lb
• Does being 8lb at birth increase the chance of
  you being born with a larger head
• Unlike the logistic model we are concerned with
  three probabilities
• P(headcirc 49.1 50.7cm)
• P(headcirc 50.8 51.9cm)
• P(headcirc 52cm)
• Each is referenced against the negative
  response i.e. that headcirc lt 49cm

37
Exposure 1 bwt gt 8lb

• . mlogit headcirc4 bwt_8lb, baseoutcome(0)
• Multinomial logistic regression
• --------------------------------------------------
  ------------
• headcirc4 Coef. SE z Pgtz
  95 CI
• -------------------------------------------------
  ------------
• 1
• bwt_8lb 1.56 .135 11.53 0.000
  1.30 1.83
• _cons -.07 .029 -2.30 0.022
  -0.12 -0.01
• -------------------------------------------------
  ------------
• 2
• bwt_8lb 3.09 .129 23.98 0.000
  2.84 3.34
• _cons -.58 .034 -17.33 0.000
  -0.65 -0.52
• -------------------------------------------------
  ------------
• 3
• bwt_8lb 4.39 .127 34.43 0.000
  4.14 4.64
• _cons -.99 .039 -25.56 0.000
  -1.06 -0.92
• --------------------------------------------------
  ------------
• (headcirc40 is the base outcome)

38
Exposure 1 bwt gt 8lb

• . mlogit headcirc4 bwt_8lb, baseoutcome(0)
• Multinomial logistic regression
• ---------------------------------
• headcirc4 Coef. (SE)
• --------------------------------
• 1
• bwt_8lb 1.56 (.135)
• _cons -.07 (.029)
• --------------------------------
• 2
• bwt_8lb 3.09 (.129)
• _cons -.58 (.034)
• --------------------------------
• 3
• bwt_8lb 4.39 (.127)
• _cons -.99 (.039)
• ---------------------------------
• (headcirc40 is the base outcome)

Logistic regression ------------------------------
---- head_1 Coef. Std.
Err. --------------------------------- bwt_8lb
1.56099 .1353772 _cons -.0664822
.0289287 ---------------------------------- Logis
tic regression ----------------------------------
head_2 Coef. Std. Err. ----------------
----------------- bwt_8lb 3.088329
.1287576 _cons -.5822197
.0335953 ---------------------------------- Logis
tic regression ----------------------------------
head_3 Coef. Std. Err. ----------------
----------------- bwt_8lb 4.389338
.127473 _cons -.9862376 .0385892 ----------
------------------------
39
Exposure 1 bwt gt 8lb

• . mlogit headcirc4 bwt_8lb, baseoutcome(0)
• Multinomial logistic regression
• ---------------------------------
• headcirc4 Coef. (SE)
• --------------------------------
• 1
• bwt_8lb 1.56 (.135)
• _cons -.07 (.029)
• --------------------------------
• 2
• bwt_8lb 3.09 (.129)
• _cons -.58 (.034)
• --------------------------------
• 3
• bwt_8lb 4.39 (.127)
• _cons -.99 (.039)
• ---------------------------------
• (headcirc40 is the base outcome)

Logistic regression ------------------------------
---- head_1 Coef. Std.
Err. --------------------------------- bwt_8lb
1.56099 .1353772 _cons -.0664822
.0289287 ---------------------------------- Logis
tic regression ----------------------------------
head_2 Coef. Std. Err. ----------------
----------------- bwt_8lb 3.088329
.1287576 _cons -.5822197
.0335953 ---------------------------------- Logis
tic regression ----------------------------------
head_3 Coef. Std. Err. ----------------
----------------- bwt_8lb 4.389338
.127473 _cons -.9862376 .0385892 ----------
------------------------
40
Exposure 1 bwt gt 8lb

• For a categorical exposure, a multinomial
  logistic model fitted over 4 outcome levels gives
  the same estimates as 3 logistic models, i.e.
• Logit(0v1)
• Multinomial(0v1,0v2,0v3) Logit(0v2)
• Logit(0v3)
• In this instance, the single model is merely more
  convenient and allows the testing of equality
  constraints

41
Exposure 2 Continuous bwt

• Using standardized birthweight we are interesting
  in how the probability of having a larger head,
  i.e.
• P(headcirc 49.1 50.7cm)
• P(headcirc 50.8 51.9cm)
• P(headcirc 52cm)
• increases as birthweight increases
• As with the binary logistic models, estimates
  will reflect
• A change in log-odds per SD change in birthweight
• The gradient or slope when in the logit scale

42
Exposure 2 Continuous bwt

• mlogit headcirc4 bwt_z, baseoutcome(0)
• Multinomial logistic regression
• --------------------------------------------------
  ------------
• headcirc4 Coef. SE z Pgtz
  95 CI
• -------------------------------------------------
  ------------
• 1
• bwt_z 2.10 .063 33.11 0.000
  1.97 2.22
• _cons 1.06 .044 23.85 0.000
  0.97 1.14
• -------------------------------------------------
  ------------
• 2
• bwt_z 3.52 .078 44.89 0.000
  3.37 3.68
• _cons 0.78 .046 16.95 0.000
  0.69 0.87
• -------------------------------------------------
  ------------
• 3
• bwt_z 4.88 .086 56.90 0.000
  4.72 5.05
• _cons 0.33 .051 6.51 0.000
  0.23 0.43
• --------------------------------------------------
  ------------
• (headcirc40 is the base outcome)

43
Exposure 2 Continuous bwt
Logistic regression ------------------------------
------- head_1 Coef. Std.
Err. ------------------------------------ bwt_z
2.093789 .0650987 _cons 1.058445
.0447811 ------------------------------------- Lo
gistic regression --------------------------------
----- head_2 Coef. Std.
Err. ------------------------------------ bwt_z
3.355041 .0959539 _cons .6853272
.0464858 ------------------------------------- Lo
gistic regression --------------------------------
----- head_3 Coef. Std.
Err. ------------------------------------ bwt_z
3.823597 .1065283 _cons .3129028
.0492469 -------------------------------------

• Multinomial logistic regression
• ------------------------------
• headcirc4 Coef. (SE)
• -----------------------------
• 1
• bwt_z 2.10 (.063)
• _cons 1.06 (.044)
• -----------------------------
• 2
• bwt_z 3.52 (.078)
• _cons 0.78 (.046)
• -----------------------------
• 3
• bwt_z 4.88 (.086)
• _cons 0.33 (.051)
• ------------------------------
• (headcirc40 is the base outcome)

No longer identical
44
Exposure 2 Continuous bwt
Outcome level 2 49.1 50.7 Intercept
1.06 Slope 2.10 Shallowest
Outcome level 3 50.8 51.9 Intercept
0.78 Slope 3.52
Outcome level 4 52.0 Intercept
0.33 Slope 4.88 Steepest Risk of being in
outcome level 4 increases most sharply as bwt
increases
45
Can plot probabilities for all 4 levels
46
Or altogether on one graph.
47
Or altogether on one graph.
48
Ordinal Logistic Models
49
Ordinal Logistic Models

• When applicable, it is useful to favour ordinal
  models over multinomial models
• If outcome levels are increasing, e.g. in
  severity of a condition or agreement with a
  statement, we expect the model parameters to
  behave in a certain way
• The typical approach is to fit ordinal models
  with constraints resulting in greater parsimony
  (less parameters)

50
Contrasting among response categories - some
alternative models
For a 4-level outcome there are three comparisons
to be made Model 1 that used in the
multinomial logistic model Model 2 used with
the proportional-odds ordinal model Model 3
adjacent category model
51
Proportional odds model

• For an ordinal outcome, all three models shown on
  the previous slide could be fitted.
• Which one is chosen will depend on the type of
  data being analysed, and the assumptions you wish
  to make
• POM is very popular in the literature (thanks to
  SPSS)
• Consists of model 2 applied with a parameter
  constraint

52
Proportional odds model
Alternative but equivalent parameterizations are
sometimes used

• The alphas do not have a subscript, hence the
  proportional
• odds assumption, AKA equal slopes

53
Proportional odds model

• As we are modelling cumulative probabilities, the
  probability of a specific response category
  occurring must be obtained by subtraction

54
E.g. for 3-category ordinal 0,1,2
55
Exposure continuous birthweight

• . gologit2 headcirc4 bwt_z, npl
• Generalized Ordered Logit Estimates
• -------------------------------
• headcirc4 Coef. (SE)
• ------------------------------
• 0
• bwt_z 2.79 (.059)
• _cons 1.90 (.040)
• ------------------------------
• 1
• bwt_z 2.61 (.049)
• _cons -0.13 (.026)
• ------------------------------
• 2
• bwt_z 2.23 (.048)
• _cons -1.55 (.034)
• -------------------------------

Unconstrained model
WARNING! 8 in-sample cases have an outcome with a
predicted probability that is less than 0. See
the gologit2 help section on Warning Messages for
more information.
56

• . gologit2 headcirc4 bwt_z if kz030gt1.25, npl
• Generalized Ordered Logit Estimates
  Number of obs 10422
• 
  LR chi2(3) 7981.91
• 
  Prob gt chi2 0.0000
• Log likelihood -10396.564
  Pseudo R2 0.2774
• --------------------------------------------------
  ----------------------------
• headcirc4 Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• 0
• bwt_z 2.785725 .0585952 47.54
  0.000 2.67088 2.900569
• _cons 1.904522 .0395861 48.11
  0.000 1.826935 1.982109
• -------------------------------------------------
  ----------------------------
• 1
• bwt_z 2.60551 .0494048 52.74
  0.000 2.508678 2.702341
• _cons -.1269783 .0260562 -4.87
  0.000 -.1780475 -.0759092
• -------------------------------------------------
  ----------------------------
• 2

57
Exposure continuous birthweight

• . gologit2 headcirc4 bwt_z, pl
• Generalized Ordered Logit Estimates
• -------------------------------
• headcirc4 Coef. (SE)
• ------------------------------
• 0
• bwt_z 2.53 (.036)
• _cons 1.78 (.031)
• ------------------------------
• 1
• bwt_z 2.53 (.036)
• _cons -0.17 (.025)
• ------------------------------
• 2
• bwt_z 2.53 (.036)
• _cons -1.70 (.030)
• -------------------------------

Constrained model
58
Exposure continuous birthweight

• Equality constraint has brought us 2 d.f.
• With more covariates (or more outcome levels) the
  savings could be considerable
• Model test
• . gologit2 headcirc4 bwt_z, npl
  store(unconstrained)
• . gologit2 headcirc4 bwt_z, pl store(constrained)
• . lrtest constrained unconstrained
• Likelihood-ratio test LR
  chi2(2) 69.78
• (Assumption constrained nested in unconstrained)
  
• Prob gt
  chi2 0.0000

59
Exposure continuous birthweight
Constrained - Proportional odds (equal slopes)
Unconstrained
60
Introduce a new model
Proportional odds model
Adjacent category model
61
Adjacent-category model

• Adjacent category model is a nominal response
• Each category in turn is compared to its nearest
  neighbour
• Within each comparison, nothing is said about the
  other categories unless model constraints are
  imposed
• Potential for testing the ordinal nature of the
  item as we will see later

62
3-category adjacent-category model
Consider a categorical outcome with 3 levels
0,1,2
63
We can rearrange this to give (us a headache)
64
And?

• Notice
• The 3 denominators are the same
• Each response category probability can be written
  DIRECTly as ratio of sums of exponents
• Also known as a divide by total method
• This is contrary to POM which was an INDIRECT or
  difference method since probabilities obtained
  through subtraction

65
Summary so far

• Aim of logistic regression is to model non-linear
  probabilistic relationship between explanatory
  variables and a categorical outcome
• The logit form of the regression is assumed
  linear in its regression terms
• Can be classified as a generalized linear model

66
Summary so far

• For polytomous outcomes, J outcome levels leads
  to J-1 regression analyses
• Appropriate (ordinal) data gives the potential
  for parsimony through constraints such as
  proportional odds

67
Worked example

• Outcome
• 4-level categorical head circumference
• Exposure
• Standardized birthweight
• Aim
• Fit the equivalent to Statas -mlogit- and
  -gologit2- functions in R and compare with
  adjacent category model

68
Ingredients

• one dataset data_for_R2.dta
• one R-script bwt headcirc for R.R
• one copy of R (latest version 2.9.2)
• R packages
• Foreign (to load dataset)
• VGAM (to run models)

69
R Multinomial / adjacent category models

• multinomial logit model - cts covariate
• fit_multinom vglm(headc4 bwtz,
  multinomial(refLevel1))
• constraints(fit_multinom)
• coef(fit_multinom, matrixTRUE)
• Summary(fit_multinom)
• 
  
• adjacent categories logit model - cts covariate
• 
  
• fit_adjcat vglm(headc4 bwtz, acat)
• constraints(fit_adjcat)
• coef(fit_adjcat, matrixTRUE)
• summary(fit_adjcat)

70
R Cumulative logistic model

• cumulative model - cts covariate
• constrained (parallel) model
• fit_pom vglm(headc4 bwtz, cumulative(parallel
  TRUE, reverseTRUE))
• constraints(fit_pom)
• coef(fit_pom, matrixTRUE)
• unconstrained (non-parallel) model
• fit_nonpom vglm(headc4 bwtz,
  cumulative(parallelFALSE, reverseTRUE))
• constraints(fit_nonpom)
• coef(fit_nonpom, matrixTRUE)
• Check the proportional odds assumption with a
  LRT
• 
  
• 1 - pchisq(2(logLik(fit_nonpom)-logLik(fit_pom)),
  dflength(coef(fit_nonpom))-length(coef(fit_pom))
  )

71
Non-POM in VGAM

• VGAM procedure struggles with unequal slopes when
  continuous predictor strongly related to outcome
• This occurred in this example and also in the
  exercises
• We are using R as a useful teaching environment,
  but it is free, and not everything is 100
  reliable

72
summary(fit_multinom)

• Call
• vglm(formula headc4 bwtz, family
  multinomial(refLevel 1))
• Coefficients
• Value Std. Error t value
• (Intercept)1 1.05708 0.044320 23.8514
• (Intercept)2 0.78024 0.046040 16.9471
• (Intercept)3 0.33206 0.050998 6.5113
• bwtz1 2.09505 0.063267 33.1146
• bwtz2 3.52171 0.078459 44.8858
• bwtz3 4.88490 0.085853 56.8987
• Number of linear predictors 3
• Names of linear predictors log(mu,2/mu,1),
  log(mu,3/mu,1), log(mu,4/mu,1)
• Dispersion Parameter for multinomial family 1
• Residual Deviance 20946.78 on 31290 degrees of
  freedom
• Log-likelihood -10473.39 on 31290 degrees of
  freedom

73
summary(fit_adjcat)

• Call
• vglm(formula headc4 bwtz, family acat)
• Coefficients
• Value Std. Error t value
• (Intercept)1 1.05708 0.044320 23.8514
• (Intercept)2 -0.27685 0.031349 -8.8312
• (Intercept)3 -0.44817 0.040190 -11.1512
• bwtz1 2.09505 0.063267 33.1146
• bwtz2 1.42666 0.057806 24.6800
• bwtz3 1.36319 0.052771 25.8323
• Number of linear predictors 3
• Names of linear predictors
• log(PY2/PY1), log(PY3/PY2),
  log(PY4/PY3)
• Dispersion Parameter for acat family 1
• Residual Deviance 20946.78 on 31290 degrees of
  freedom

74
summary(fit_pom)

• Call
• vglm(formula headc4 bwtz, family
  cumulative(parallel TRUE,
• reverse TRUE))
• Pearson Residuals
• Min 1Q Median
  3Q Max
• logit(PYgt2) -20.3575 0.013626 0.138778
  0.46234 8.1378
• logit(PYgt3) -9.7316 -0.461336 0.030467
  0.44740 12.7044
• logit(PYgt4) -6.0701 -0.374240 -0.134896
  0.30254 14.7435
• Coefficients
• Value Std. Error t value
• (Intercept)1 1.78467 0.031368 56.8950
• (Intercept)2 -0.17112 0.025049 -6.8316
• (Intercept)3 -1.70386 0.030295 -56.2419
• bwtz 2.52516 0.036331 69.5052
• Number of linear predictors 3
• Names of linear predictors logit(PYgt2),
  logit(PYgt3), logit(PYgt4)

75
summary(fit_nonpom)

• Call
• vglm(formula headc4 bwtz, family
  cumulative(parallel FALSE,
• reverse TRUE), maxit 100)
• Pearson Residuals
• Min 1Q Median
  3Q Max
• logit(PYgt2) -26.8061 0.010352 0.127186
  0.39716 10.639
• logit(PYgt3) -10.2091 -0.488329 0.031261
  0.44992 15.139
• logit(PYgt4) -4.8869 -0.375265 -0.151709
  0.36328 10.577
• Coefficients
• Value Std. Error t value
• (Intercept)1 1.90529 0.0392664 48.5222
• (Intercept)2 -0.12164 0.0238242 -5.1056
• (Intercept)3 -1.54436 0.0249593 -61.8751
• bwtz1 2.77653 0.0566026 49.0531
• bwtz2 2.57170 0.0085618 300.3694
• bwtz3 2.21528 0.0048085 460.7037

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Summary of estimates
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Plots for multinomial model
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Plots for adjacent category model
LOGITS have different interpretation compared
with last model
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Plots for cumulative models POM
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OCCs for POM
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Plots for cumulative models non-POM
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OCCs for non-POM
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Some decision about which is best?

• Seen an exhausting but non-exhaustive range of
  ordinal models
• The decision regarding which is best is not
  really a statistical one
• One should always consider a variety of models to
  assess whether ones conclusions are affected by a
  (perhaps arbitrary) model choice
• Here practical differences are likely to be
  minor, at least away from the distribution tails

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Now for some exercises
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Exercise 1 binary predictor

• Categorical outcome and binary predictor
• Using a simple crosstab, estimate log-odds
  estimates for
• Multinomial logistic
• Un-equal slopes cumulative logistic
• Adjacent category logistic
• Verify the results by fitting the equivalent
  models in R using -VGAM-

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Exercise 2 continuous predictor

• Categorical outcome and continuous predictor
• Using the -VGAM- function estimate the following
  models
• Multinomial logistic
• Un-equal slopes cumulative logistic
• Equal slopes cumulative logistic (POM)
• Adjacent category logistic
• Examine the predicted probabilities/logits
  graphically for each model

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Data for the exercise bmi11to14data.dta-

• Binary predictor (bmi7_med)
• BMI median (15.6) at age 7
• Continuous predictor (bmi7/bmi7z)
• cts BMI at age 7
• 4-level categorical outcome (bmi14_g4)
• BMI at age 14
• Categorised min/200 20/221 22/242 24/max3

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Exercise 1

• Crosstab of bmi7gt20 and bmi14_g4
• bmi14_g4
• bmi7_med 0 1 2
  3 Total
• -------------------------------------------------
  ----------------
• 0 2,079 313 71
  37 2,500
• 83.2 12.5 2.8
  1.5
• -------------------------------------------------
  ----------------
• 1 707 690 464
  639 2,500
• 28.3 27.6 18.6
  25.6
• -------------------------------------------------
  ----------------
• Total 2,786 1,003 535
  676 5,000

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Optional homework

• Refit these models in a more familiar stats
  package e.g.
• SPSS nomreg / PLUM
• (http//www.norusis.com/pdf/ASPC_v13.pdf)
• Stata mlogit / gologit2
• http//www.ats.ucla.edu/stat/Stata/dae/ologit.htm
• SAS proc logistic
• http//www.ats.ucla.edu/stat/sas/dae/ologit.htm
• ACAT not possible in SPSS, nor Stata (except
  possibly in GLLAMM)

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http//cran.r-project.org/
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