Bug in Stata’s dendrogram code

Binary treeDendrograms are diagrams that have a tree-like structure, and they’re often used to represent the structure of clustering in a hierarchical (agglomerative) cluster analysis. Agglomerative clustering starts from the bottom up, joining the nearest pairs of objects into clusters, and then clusters with objects and finally clusters with clusters, until eventually everything is a single cluster. The single cluster is the root, the objects are the leaves, and in between is a binary tree, where objects and clusters are combined depending on their distance from each other.

This process depends on being able to define a distance between an object and a cluster, and between pairs of clusters, and there are various ways to do this. However, some algorithms may cause the distance between an object/cluster and another cluster to change after the amalgamation of other clusters. This permits “reversals”, which are difficult or impossible to represent in a dendrogram-like structure. But clustering algorithms or “linkages” such as Ward’s are not subject to this problem.

OK, so far so good. What’s the problem? Stata’s dendrogram code is slightly buggy, and can give an error:
currently can't handle dendrogram reversals
even when you are using a linkage that is not subject to reversals. The explanation is that it is comparing distances between pairs of clusters where one must be greater than or equal to the other for the dendrogram to be drawable (otherwise it’s a reversal), and due to numeric precision is finding pairs where one is fractionally less. The correct code should test the difference in the distances is not less than a very small number (e.g., 10^-7) to take account of precision.

I have had a number of people report this error to me, in connection with my SADI Stata ado package, and have been able to reproduce it. In cooperation with some of these respondents, we have been able to get a work-around from Stata Technical Support (this bug persists into Stata 14).

The following command displays the information that Stata holds about the current clustering (apologies for the linewrapping):

. char list _dta[_clus_1_info]
_dta[_clus_1_info]: `"t hierarchical"' `"m wards"' `"d user matrix omd"' `"v id _clus_1_id"' `"v order _clus_1_ord"' `"v height _clus_1_hgt"'

A small change to this will allow the dendrogram to be drawn:

. char _dta[_clus_1_info] `"`"t hierarchical"' `"m wards"' `"d user matrix omd"' `"v id _clus_1_id"' `"v order _clus_1_ord"' `"v height _clus_1_hgt"'"'

Gender change in Irish political representation

Estimating the effect of the gender quota: The initial question

A gender quota for candidates was imposed in the 2016 Irish election (see e.g., http://www.thejournal.ie/readme/gender-quota). The question arises whether it had an impact. As a first pass consider the following table:1

           |        gender
      dail |         f          m |     Total
      2011 |        21        145 |       166 
           |     12.65      87.35 |    100.00 
      2016 |        32        116 |       148 
           |     21.62      78.38 |    100.00 
     Total |        53        261 |       314 
           |     16.88      83.12 |    100.00

This is clearly a big rise. Is it statistically significant?

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Pseudo-R2 is pseudo

People like the R2 stat from linear regression so much that they re-invent it in places it doesn’t naturally arise, such as logistic regression. The true R2 has nice clean interpretations, as the proportion of variation explained or the square of the correlation between observed and predicted values. The fake or pseudo-R2 statistics are often based on relating the loglikelihood of the current model against that of the null model (intercept only) in some way. There is a good overview at UCLA.

One of the most popular pseudo-R2 is McFadden’s. This is defined as 1 – LLm/LL0 where LLm is the log-likelihood of the current model, and LL0 that of the null model. This appears to have the range 0-1 though 1 will never be reached in practice.

It is well known that if we fit linear regressions by maximum-likelihood, we get exactly the same parameter estimates as if we fit by ordinary least squares. We can demonstrate this in Stata:

. sysuse auto
. reg price headroom mpg
. glm price headroom mpg

Since the ML estimation of the linear regression gives us loglikelihoods, we can calculate pseudo-R2 and true R2 for the same model. This code does it for a range of simple models with Stata’s demonstration “auto” data set:

sysuse auto, clear
glm price
local basell = e(ll)
local vars "mpg rep78 headroom trunk weight length turn displacement gear_ratio foreign"
local rhs = ""

gen r2 = .
gen mcf = .

local i 0
foreach var in `vars' {
local i = `i'+1
local rhs = "`rhs' `var'"
qui glm price `rhs'
local mcfad = 1 - (e(ll)/ `basell')
qui reg price `rhs'
di %6.3f `=e(r2)' %6.3f `mcfad' " : `rhs'"
qui replace r2 = `=e(r2)' in `i'
qui replace mcf = `mcfad' in `i'

label var mcf "McFadden Pseudo-R2"
label var r2 "R-squared"
scatter mcf r2

This generates the following graph, in which we see that there is a monotonic but non-linear relationship between the two measures. We can also see very clearly that pseudo-R2 is always substantially lower than R2. Thus it should be clear that while it emulates R2 in spirit, it doesn’t actually approximate it. So when people talk about proportion of variation explained in a logistic regression, shoot them down.

pseudo-R2 vs R2

UCAS, ethnicity and admission rates

UCAS, the UK university admissions clearing house, have released data relating to ethnicity and admissions to English universities, in part in response to Vikki Boliver‘s research in Sociology suggesting that members of ethnic minorities are less likely to be admitted to Russell Group universities.

The analysis note with the release is sober and correct, showing a mostly consistent pattern of offer rates for ethnic minority students being lower (but not far lower) than expected. However, UCAS’s press release seems to have suggested that the effect is almost explained away, and attributes it to ethnic minority students disproportionately applying to courses with low acceptance rates. This does not seem to be the case.

Update: see also next blog entry.
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Substitution costs from transition rates

Given that determining substitution costs in sequence analysis is such a bone of contention, many researchers look for a way for the data to generate the costs. The typical way to do this is, is by pooling transition rates and defining the substitution cost to be:

2 – p(ij) – p(ji)

where p(ij) is the transition rate from state i to state j. Intuitively, states that are closer to each other will have higher transitions, and vice versa. Continue reading