You can use the loglm() function in the
MASS package to fit log-linear models. Equivalent models
can also be fit (from a different perspective) as generalized linear
models with the glm() function using the
family='poisson' argument, and the gnm package
provides a wider range of generalized nonlinear models,
particularly for testing structured associations.
The visualization methods for these models were originally developed
for models fit using loglm(), so this approach is
emphasized here. Some extensions of these methods for models fit using
glm() and gnm() are contained in the
vcdExtra package and illustrated in (ref?)(sec:glm).
Assume we have a 3-way contingency table based on variables A, B, and
C. The possible different forms of loglinear models for a 3-way table
are shown in the table below. @(tab:loglin-3way) The Model
formula column shows how to express each model for
loglm() in R. 1 In the Interpretation
column, the symbol “\(\perp\)” is to be
read as “is independent of,” and “\(\;|\;\)” means “conditional on,” or
“adjusting for,” or just “given”.
| Model | Model formula | Symbol | Interpretation |
|---|---|---|---|
| Mutual independence | ~A + B + C |
\([A][B][C]\) | \(A \perp B \perp C\) |
| Joint independence | ~A*B + C |
\([AB][C]\) | \((A \: B) \perp C\) |
| Conditional independence | ~(A+B)*C |
\([AC][BC]\) | \((A \perp B) \;|\; C\) |
| All two-way associations | ~A*B + A*C + B*C |
\([AB][AC][BC]\) | homogeneous association |
| Saturated model | ~A*B*C |
\([ABC]\) | 3-way association |
For example, the formula ~A + B + C specifies the model
of mutual independence with no associations among the three
factors. In standard notation for the expected frequencies \(m_{ijk}\), this corresponds to
\[ \log ( m_{ijk} ) = \mu + \lambda_i^A + \lambda_j^B + \lambda_k^C \equiv A + B + C \]
The parameters \(\lambda_i^A , \lambda_j^B\) and \(\lambda_k^C\) pertain to the differences among the one-way marginal frequencies for the factors A, B and C.
Similarly, the model of joint independence, \((A \: B) \perp C\), allows an association between A and B, but specifies that C is independent of both of these and their combinations,
\[ \log ( m_{ijk} ) = \mu + \lambda_i^A + \lambda_j^B + \lambda_k^C + \lambda_{ij}^{AB} \equiv A * B + C \]
where the parameters \(\lambda_{ij}^{AB}\) pertain to the overall association between A and B (collapsing over C).
In the literature or text books, you will often find these models
expressed in shorthand symbolic notation, using brackets,
[ ] to enclose the high-order terms in the model.
Thus, the joint independence model can be denoted [AB][C],
as shown in the Symbol column in the table.
@(tab:loglin-3way).
Models of conditional independence allow (and fit) two of
the three possible two-way associations. There are three such models,
depending on which variable is conditioned upon. For a given conditional
independence model, e.g., [AB][AC], the given variable is
the one common to all terms, so this example has the interpretation
\((B \perp C) \;|\; A\).
loglm()For example, we can fit the model of mutual independence among hair
color, eye color and sex in HairEyeColor as
library(MASS)
## Independence model of hair and eye color and sex.
hec.1 <- loglm(~Hair+Eye+Sex, data=HairEyeColor)
hec.1
## Call:
## loglm(formula = ~Hair + Eye + Sex, data = HairEyeColor)
##
## Statistics:
## X^2 df P(> X^2)
## Likelihood Ratio 166.3001 24 0
## Pearson 164.9247 24 0Similarly, the models of conditional independence and joint independence are specified as
## Conditional independence
hec.2 <- loglm(~(Hair + Eye) * Sex, data=HairEyeColor)
hec.2
## Call:
## loglm(formula = ~(Hair + Eye) * Sex, data = HairEyeColor)
##
## Statistics:
## X^2 df P(> X^2)
## Likelihood Ratio 156.6779 18 0
## Pearson 147.9440 18 0## Joint independence model.
hec.3 <- loglm(~Hair*Eye + Sex, data=HairEyeColor)
hec.3
## Call:
## loglm(formula = ~Hair * Eye + Sex, data = HairEyeColor)
##
## Statistics:
## X^2 df P(> X^2)
## Likelihood Ratio 19.85656 15 0.1775045
## Pearson 19.56712 15 0.1891745Note that printing the model gives a brief summary of the goodness of
fit. A set of models can be compared using the anova()
function.
anova(hec.1, hec.2, hec.3)
## LR tests for hierarchical log-linear models
##
## Model 1:
## ~Hair + Eye + Sex
## Model 2:
## ~(Hair + Eye) * Sex
## Model 3:
## ~Hair * Eye + Sex
##
## Deviance df Delta(Dev) Delta(df) P(> Delta(Dev)
## Model 1 166.30014 24
## Model 2 156.67789 18 9.62225 6 0.14149
## Model 3 19.85656 15 136.82133 3 0.00000
## Saturated 0.00000 0 19.85656 15 0.17750glm() and gnm()The glm() approach, and extensions of this in the
gnm package allows a much wider class of models for
frequency data to be fit than can be handled by loglm(). Of
particular importance are models for ordinal factors and for square
tables, where we can test more structured hypotheses about the patterns
of association than are provided in the tests of general association
under loglm(). These are similar in spirit to the
non-parametric CMH tests described in @ref(sec:CMH).
Example: The data Mental in
the vcdExtra package gives a two-way table in frequency
form classifying young people by their mental health status and parents’
socioeconomic status (SES), where both of these variables are ordered
factors.
data(Mental, package = "vcdExtra")
str(Mental)
## 'data.frame': 24 obs. of 3 variables:
## $ ses : Ord.factor w/ 6 levels "1"<"2"<"3"<"4"<..: 1 1 1 1 2 2 2 2 3 3 ...
## $ mental: Ord.factor w/ 4 levels "Well"<"Mild"<..: 1 2 3 4 1 2 3 4 1 2 ...
## $ Freq : int 64 94 58 46 57 94 54 40 57 105 ...
xtabs(Freq ~ mental + ses, data=Mental) # display the frequency table
## ses
## mental 1 2 3 4 5 6
## Well 64 57 57 72 36 21
## Mild 94 94 105 141 97 71
## Moderate 58 54 65 77 54 54
## Impaired 46 40 60 94 78 71Simple ways of handling ordinal variables involve assigning scores to
the table categories, and the simplest cases are to use integer scores,
either for the row variable
(column effects'' model), the column variable (row
effects’’ model), or both (``uniform association’’ model).
To fit more parsimonious models than general association, we can define numeric scores for the row and column categories
# Use integer scores for rows/cols
Cscore <- as.numeric(Mental$ses)
Rscore <- as.numeric(Mental$mental) Then, the row effects model, the column effects model, and the uniform association model can be fit as follows. The essential idea is to replace a factor variable with its numeric equivalent in the model formula for the association term.
# column effects model (ses)
coleff <- glm(Freq ~ mental + ses + Rscore:ses, family = poisson, data = Mental)
# row effects model (mental)
roweff <- glm(Freq ~ mental + ses + mental:Cscore, family = poisson, data = Mental)
# linear x linear association
linlin <- glm(Freq ~ mental + ses + Rscore:Cscore, family = poisson, data = Mental)The LRstats() function in vcdExtra provides
a nice, compact summary of the fit statistics for a set of models,
collected into a glmlist object. Smaller is better for AIC and
BIC.
# compare models using AIC, BIC, etc
vcdExtra::LRstats(glmlist(indep, roweff, coleff, linlin))
## Likelihood summary table:
## AIC BIC LR Chisq Df Pr(>Chisq)
## indep 209.59 220.19 47.418 15 3.155e-05 ***
## roweff 174.45 188.59 6.281 12 0.9013
## coleff 179.00 195.50 6.829 10 0.7415
## linlin 174.07 185.85 9.895 14 0.7698
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1For specific model comparisons, we can also carry out tests of
nested models with anova() when those models are
listed from smallest to largest. Here, there are two separate paths from
the most restrictive (independence) model through the model of uniform
association, to those that allow only one of row effects or column
effects.
anova(indep, linlin, coleff, test="Chisq")
## Analysis of Deviance Table
##
## Model 1: Freq ~ mental + ses
## Model 2: Freq ~ mental + ses + Rscore:Cscore
## Model 3: Freq ~ mental + ses + Rscore:ses
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 15 47.418
## 2 14 9.895 1 37.523 9.035e-10 ***
## 3 10 6.829 4 3.066 0.5469
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(indep, linlin, roweff, test="Chisq")
## Analysis of Deviance Table
##
## Model 1: Freq ~ mental + ses
## Model 2: Freq ~ mental + ses + Rscore:Cscore
## Model 3: Freq ~ mental + ses + mental:Cscore
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 15 47.418
## 2 14 9.895 1 37.523 9.035e-10 ***
## 3 12 6.281 2 3.614 0.1641
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The model of linear by linear association seems best on all accounts. For comparison, one might try the CMH tests on these data:
CMHtest(xtabs(Freq~ses+mental, data=Mental))
## Cochran-Mantel-Haenszel Statistics for ses by mental
##
## AltHypothesis Chisq Df Prob
## cor Nonzero correlation 37.156 1 1.0907e-09
## rmeans Row mean scores differ 40.297 5 1.3012e-07
## cmeans Col mean scores differ 40.666 3 7.6971e-09
## general General association 45.958 15 5.4003e-05The strength of the gnm package is that it handles a
wide variety of models that handle non-linear terms, where the
parameters enter the model beyond a simple linear function. The simplest
example is the Goodman RC(1) model (Goodman,
1979), which allows a multiplicative term to account for the
association of the table variables. In the notation of generalized
linear models with a log link, this can be expressed as
\[ \log \mu_{ij} = \alpha_i + \beta_j + \gamma_{i} \delta_{j} ,\]
where the row-multiplicative effect parameters \(\gamma_i\) and corresponding column parameters \(\delta_j\) are estimated from the data.% 2
Similarly, the RC(2) model adds two multiplicative terms to the independence model,
\[ \log \mu_{ij} = \alpha_i + \beta_j + \gamma_{i1} \delta_{j1} + \gamma_{i2} \delta_{j2} . \]
In the gnm package, these models may be fit using the
Mult() to specify the multiplicative term, and
instances() to specify several such terms.
Example: For the Mental data,
we fit the RC(1) and RC(2) models, and compare these with the
independence model.
RC1 <- gnm(Freq ~ mental + ses + Mult(mental,ses), data=Mental,
family=poisson, verbose=FALSE)
RC2 <- gnm(Freq ~ mental+ses + instances(Mult(mental,ses),2), data=Mental,
family=poisson, verbose=FALSE)
anova(indep, RC1, RC2, test="Chisq")
## Analysis of Deviance Table
##
## Model 1: Freq ~ mental + ses
## Model 2: Freq ~ mental + ses + Mult(mental, ses)
## Model 3: Freq ~ mental + ses + Mult(mental, ses, inst = 1) + Mult(mental,
## ses, inst = 2)
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 15 47.418
## 2 9 40.230 6 7.188 0.3038
## 3 3 0.523 6 39.707 5.2e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1For glm(), or gnm(), with the
data in the form of a frequency data.frame, the same model is specified
in the form glm(Freq \(\sim\) ..., family="poisson"),
where Freq is the name of the cell frequency variable and
... specifies the Model formula.↩︎
This is similar in spirit to a correspondence analysis with a single dimension, but as a statistical model.↩︎