An alternative way to interpret the ordinal model is with cumulative probabilities. I don’t find these as helpful as predicted probabilities, but the basic idea is important to understand for odds ratios interpretations (see Long and Freese 2014 for more on cumulative probability interpretations and plots)
For a unit increase in xk the odds of higher outcomes compared to lower outcomes change by a factor of\(\exp(\beta_{k}^{*})\), holding other variables constant
Odds ratios interpretations in ordinal models
The odds are identical regardless where you binarize the comparison:
The odds ratio is interpreted in terms of the underlying latent variable (y*)
Effective interpretations are contingent on being able to properly and clearly name and talk about this underlying latent variable
Remember this must be a single unidirectional latent concept to satisfy the assumptions of the model
If you cannot easily or effectively decide what the latent variable is this is a pretty good indicator an ordinal model might not be appropriate
Naming the latent variable y*
How would you name and talk about the latent variable y* if you used one of the following ordinal variables as a DV?
All examples from the General Social Survey (GSS) which mostly consists of questions asked as ordinal variables
Would you say your own health, in general, is excellent, good, fair, or poor?
How often do you read the newspaper — every day, a few times a week, once a week, less than once a week, or never?
Because of past discrimination, employers should make special efforts to hire and promote qualified women…please tell me whether you strongly agree, agree, neither agree nor disagree, disagree, or strongly disagree.
Methods of birth control should be available to teenagers between the ages of 14 and 16 even if their parents do not approve… strongly agree, agree, disagree, or strongly disagree.
Good practice is to code your dependent variable in a way that will aid interpretation: higher numbers should = more of the latent construct
Odds ratios interpretations in ordinal models
I reverse coded the DV so that higher #s indicate more agreement for more natural interpretations
For these interpretations I am using terminology for the latent variable y* for the DV “less to more supportive of access to birth control:”
Each year since this question was first asked, the odds of being more supportive has increased by a factor of 1.006
A year increase in age decreases the odds of supporting birth control for teenagers by a factor of about 0.98
Women’s odds of having supportive attitudes for teenage birth control access are higher than men’s by a factor of about 1.38
Religious individuals’ odds of supporting teenage birth control access is 0.41 times lower than non-religious individuals
Interpretation tips: As with binary models
Need to make sure to use word “odds” in interpretation so metric is clear
Need to indicate effect is multiplicative with “times” or “by a factor”, or similar
Pros and cons of ORs in ordinal models
ORs have some appealing properties in the ordinal model
There is only one for each IV
Fewer than the number of MEs to examine
The effect is the same across the data space—i.e., you can just say “holding all else constant”
MEs will vary based on where in the data you calculate them as Pr(y = j) is nonlinear
If you are interested in the latent variable itself, ORs are in this metric
Limitations of ORs
There is no information about which categories are more or less affected*
Even if the OR is significant this does not necessarily mean that a ME for each outcome category will be significant
The substantive size of effects are not clear from the OR
You cannot compare ORs across models due to rescaling
The effects are multiplicative; not additive
*Note that when you estimate a linear regression for an ordinal outcome your β estimates are similarly in terms of the latent variable, but with the additional (usually unreasonable) assumption of meaningful and equidistance spacing between categories
Standardized coefficients
A brief primer (or reminder) of what standardized coefficients are in the linear regression context
Standardized coefficients convert the metric of the coefficient (β) into standard deviation units
We can x-standardize so the coefficient is in terms of a standard deviation increase in x
We can y-standardize so the coefficient is in terms of a standard deviation of y
We can fully (xy) standardize (apply both x and y standardization)
Note that we only x-standardize continuous IVs
The SD of a binary/nominal IV is solely a function of the mean, so not meaningful
A standard deviation increase in education completed (about 3 years), is associated with a 0.34 standard deviation increase in hourly wages
Interpreting ordinal models with y* standardization
With ordinal models, a y*-standardized interpretation can often be a satisfactory alternative to linear regression if your interest is primarily in interpreting effects on the underlying latent variable y*
You also have the ability to examine the probability of specific categories
Interpretation: for a unit increase in\(x_{k}\),\(y^{*}\)is expected to increase by\(\beta_{k}^{Sy^{*}}\)standard deviations holding other variables constant
y* standardization of the coefficients
To y* standardize the coefficients we need to calculate the variance of y*
Where the variance of the errors [\(Var(\varepsilon)\)] is assumed to be 1 in probit; \(\pi^{2}/3\) in logit
The \(\widehat{\boldsymbol{\beta}}'\widehat{Var}(\mathbf{x})\widehat{\boldsymbol{\beta}}\) component* represents the variance in the predicted y* metric
We could calculate this by making predictions for everyone in the sample in the \(\mathbf{x}\widehat{\boldsymbol{\beta}}\) metric and then calculating the variance of these predictions
Or, can calculate it via matrix algebra as formula above shows
*Note: \(\widehat{\boldsymbol{\beta}}\) is a vector of regression coefficients and that the \(\tau\)’s are not included. \(\widehat{Var}(\mathbf{x})\) is the covariance matrix of the x variables
# x, y*, and xy standardized coefficients in one tableb <-coef(pillmod)sdx <-apply(model.matrix(pillmod)[, names(b)], 2, sd) # SD of each xround(data.frame(b, bStdX = b * sdx, bStdY = b / sdystar,bStdXY = b * sdx / sdystar, SDofX = sdx), 3)
*See Appendix A5.3 for how to plot standardized coefficients in Stata and R
Women are about 0.16 standard deviations more supportive of teenage birth control, compared to men
A college degree increases support for teenage access to birth control by about 0.13 standard deviations (p < 0.001)
A standard deviation increase in age (about 17 years) decreases support for teenage birth control access by about 0.18 standard deviations.
Potential drawbacks of coefficient interpretations
The metric of y* is determined mainly by the thresholds (\(\tau\))
Instead of assuming equidistant category spacing, the ordinal model spaces the outcome categories based on the \(\tau\) estimates
How do we estimate what the thresholds are?
To start, the only information we have is the proportion of responses in each outcome category (the Percent column of the pillokR frequency table above)
Estimating thresholds in the ordinal model
Recall that the ordinal model is a cumulative model in its design and coefficients. So, we examine the cumulative proportion in each category (proportion ≤ j; the Cum. column of the frequency table)
If we assume an underlying standardized normal distribution [N ~ (0,1)] for y*, convert the cumulative proportions to z-scores:
# Show estimated thresholds (taus)probmod <-polr(pillokR ~1, data = gss, method ="probit", Hess =TRUE)summary(probmod)
Call:
polr(formula = pillokR ~ 1, data = gss, Hess = TRUE, method = "probit")
No coefficients
Intercepts:
Value Std. Error t value
StrDisag|Disagree -0.9299 0.0093 -99.8995
Disagree|Agree -0.2539 0.0080 -31.6514
Agree|StrAgree 0.5787 0.0084 68.6503
Residual Deviance: 68085.65
AIC: 68091.65
Thus, the metric of y* is determined partially by the distribution of the dependent variable
Do you find this a substantively meaningful way to decide on the category spacing of the latent continuum y*?
Can you think of a better alternative?
The good and the bad of the thresholds
The reason for setting the spacing of the thresholds based on the cumulative distribution is clear: it allows the model to uncover probabilities whose distribution match those of the observed proportions in the data (compare to the pillokR frequency table above)
![[5.3.c] Standard normal density of a hypothetical y-star split by dashed lines at tau 1 = -0.930, tau 2 = -0.254, and tau 3 = 0.579 into four shaded areas: category 1 (17.6%), 2 (22.4%), 3 (31.9%), and 4 (28.1%).](fig/cda-05_03_c-ystar.png)
![[5.3.c] Standard normal density of a hypothetical y-star split by dashed lines at tau 1 = -0.930, tau 2 = -0.254, and tau 3 = 0.579 into four shaded areas: category 1 (17.6%), 2 (22.4%), 3 (31.9%), and 4 (28.1%).](5-3-coefficients_files/figure-html/unnamed-chunk-8-1.png)