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I have data sets of the attendance of congresspeople, and of the weather in Washington, DC. I would like to examine whether and to what degree precipitation is correlated with the likelihood of individual congresspeople being present or absent.

For example, the 111th congress was in session 298 days of the 388 days between 2009-01-06 and 2010-01-29. Precipitation was recorded during 187 of those 388 days. (I also have the amount of precipitation, the temperature, the pressure, lots of weather measurements)

Congresswoman A was absent 52 times; of those 52, there was precipitation 39 times. Et cetera for each congressperson.

The null hypothesis would be that there is no relationship between precipitation and attendance.

I'd like to create some sort of visualization; perhaps a histogram of number of congresspeople on the y axis and... er... something Bayesian? Chances they will be absent, given the fact it's raining?

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  • $\begingroup$ I think you have to ask the question in a more strict way such as "is the number of attendants to congress in rainy days significantly different from those in non-rainy days"? For me the most straightforward question is: is the number of people attending significantly different when it's not raining from when it is raining? But given the nature of your dataset it looks like the cquestion can be much more complicated. $\endgroup$ Jul 7, 2014 at 15:35

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whether and to what degree precipitation is correlated with the likelihood of individual congresspeople being present or absent.

There are several options to do this through looking at correlations. One way would be to theorize that underneath both binary variables are latent normal random variables. For example, the attendance latent variable could correspond to willpower to attend projected to $\mathbb{R}$. Then you could find the correlation between these two latent variables in the normal sense. This is the method of polychoric correlation. The R package polycor does this. An example taken from the manual:

set.seed(12345)
data <- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))
x <- data[,1]
y <- data[,2]
cor(x, y) # sample correlation
x <- cut(x, c(-Inf, .75, Inf))
y <- cut(y, c(-Inf, -1, .5, 1.5, Inf))
polychor(x, y, ML=TRUE, std.err=TRUE)

#Polychoric Correlation, ML est. = 0.5231 (0.03819)
#Test of bivariate normality: Chisquare = 2.739, df = 2, p = 0.2543
#
#  Row Threshold
#  Threshold Std.Err.
#   0.7537  0.04403
#   
#  Column Thresholds
#  Threshold Std.Err.
#1   -0.9842  0.04746
#2    0.4841  0.04127
#3    1.5010  0.06118

You can see it gave you (1) the ML estimate of the correlation with standard error and (2) the threshold estimates for the latent standard normal random variable and their standard errors. This is one approach.


The null hypothesis would be that there is no relationship between precipitation and attendance.

You could test this hypothesis using the correlation above, or you could take a more comprehensive approach and include the other controls you mentioned (amount of precipitation, the temperature, the pressure, lots of weather measurements). This can be accomplished through logistic regression, probit regression, or any other binary regression approach which gives you an understanding of error.

For example, here's some fake data made in R:

set.seed(1932403)
rain = rbinom(1000, 1, 0.2)
rain = (rain == 1)
temp = rnorm(1000)
z = 3 + -2*rain + -1*temp
pr = 1/(1+exp(-z))      
attend = rbinom(1000,1,pr)
df = data.frame(attend=attend, rain=rain, temp=temp)
head(df)

#  attend  rain       temp
#1      1  TRUE -1.5418903
#2      1 FALSE  0.2950457
#3      1 FALSE -1.0759618
#4      1 FALSE  0.1218176
#5      1 FALSE -1.1599080
#6      0 FALSE  0.8309843

table(rain, attend)/1000
#       attend
#rain      0   1
#  FALSE  42 757
#  TRUE   63 138

So we can build a logistic model to test if the precipitation is important after controlling for temperature.

m = glm( attend ~ rain + temp, data=df, family="binomial")
summary(m)

#Coefficients:
#            Estimate Std. Error z value Pr(>|z|)    
#(Intercept)   3.3835     0.2003  16.890  < 2e-16 ***
#rainTRUE     -2.5190     0.2508 -10.043  < 2e-16 ***
#temp         -1.0918     0.1367  -7.985 1.41e-15 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

And in this fake data it is.


Chances they will be absent, given the fact it's raining?

You can plot (and of course, otherwise examine) the predicted probabilities using the model we just built. That is, you can look at the predicted probabilities under different conditions* (here 'TRUE' refers to rain being present):

library(lattice)
xyplot(predict(m, df, type = 'response') ~ df$temp | factor(df$rain), ylab = 'Predicted probability of attendance', xlab = 'Temperature')

enter image description here

Taking a Bayesian approach won't alter the framework you're trying to achieve: predict/build a statistical model around the probability of attendance given certain conditions (rain, temperature, etc.). There are lots of reasons to consider a Bayesian analysis, but none of them stand out as specific to your exact problem.

Another approach (naive to other controls like temperature) would be to do a proportion test.

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