# census statistical techniques

All of my statistical training has been about dealing with samples (psych background). I am involved in a project where we have a census dataset (demographic data from all the relocation sites throughout Phnom Penh, Cambodia). I am not certain if the usual methods of comparing group means, such as using an ANOVA or Kruskal-Wallis test, apply to census data. For example, we want to know whether older relocation sites are associated with a higher percentage of households with toilets. If the assumptions for normality/homoscedasticity etc are met I'd do a Pearson correlation. There are issues with the distributions of both variables though, so we're using Kendall's τ.

The percent toilets variable is severely negatively skewed (most sites have 100% households with toilets) and no amount of transformation would make the distribution normal (I tried square root, log and inverse and reflect transformations although i don't know how to do a Box Cox). Hence we decided to try splitting it up into three groups, few HH with toilets, a moderate percentage with toilets, and most HH with toilets. With census data, I suspect I should not use ANOVA/parametric equivalent to test the significance of the difference between group means. Do I just report the group means and comment on the difference/lack of difference between them?

Thanks. Any references for census statistical analysis appreciated. I have Andy Field's SPSS text book but it's all about samples, which is a pity. I've been Googling all day...

When you have a sample you use inferential stats to generalise to the population. When you have a census you already have data for the whole population, so there is no need to generalise.

For example, if you used sampling, and there is a 3% difference between groups, then you have to use inferential stats to decide whether that 3% difference is real, or just due to random chance when you did the sampling.

But if you did a census, and there is a 3% difference between groups, well, then there's definitely a 3% difference. That 3% difference is not due to random chance in sampling, because you have data for the whole population. However, even with a census you will still need to use your own judgement to think about why there is a 3% difference (for reasons other than random chance in sampling), and whether the 3% difference is large enough to have any practical significance for the work you are doing.

So basically, just use descriptive stats. Correlations are fine, but you only need the r value to show the strength of the correlation, not the p value which is related to random chance in sampling.

A lot of people don't get the difference between sample stats and census stats, and will complain that you didn't do the stats properly. I've had cases where I ended up having to do inferential stats on census data just because people complained so much that there were no p values on anything!

If you have a lot of missing data from a census sometimes you need some fancy inferential stats to fill it in. I doubt this will apply to you, but it does apply to the US population census because (for some bizarre libertarian reason) completing the census survey in not mandatory in the US.

• If you have census data, then you have population means (and all other properties of the distribution) for variables (at that point in time, at least), so there is no estimation problem - for example, the null hypothesis of no difference in means is only true when the observed means are exactly the same. For statistical analysis of census data to make sense, you have to be conceiving of your data as realizations from some hypothetical infinite population whose parameter values you want to estimate. Commented Apr 27, 2012 at 4:19
• I support Macro's comment, but would add that conceiving your data that way can often be a meaningful thing to do eg if you are interested for evaluation or policy purposes on the impact of relocation on toilets. Commented Apr 27, 2012 at 5:57
• My big question is: what happens in the in-between case where you end up with a very large sample (say 50% of the population)? Assuming there are no sampling/non-response issues. Do you use inferential stats and know that your intervals are very conservative? Commented Jun 7, 2012 at 15:46
• @Wayne Then you need a FPC (finite population correction). Commented Aug 21, 2014 at 23:25
• Does anyone have a reference for applying a FPC with a multivariate method, say OLS regression? Commented Jul 12, 2017 at 16:02

I think your bigger issue is actually not census v. sample (and for that, see my comment) but the appropriate way to compare proportions. I'd drop any idea of approximating to normal and use logistic regression, treating the households as trials and those with toilets as a success.

Breaking your nice proportion data into categories is a shame as you lose a lot of information.

• I disagree with Peter and agree with @Macro, and I'll restate Macro's point more categorically: inferential statistics are for the purpose of generalizing from a sample to a population, and if you already know the results for the population, it is of no use (and makes no sense) to conduct an inferential test such as an ANOVA, a Wald test in logistic regression, etc. Anything that generates a p-value is going to be out of place here. What I'd recommend is to explore your possibilities for effective data visualization--in a purely descriptive, not inferential, spirit. Commented Apr 27, 2012 at 12:04
• @rolando2 the population/sample way of thinking about statistics is one approach and an easy way to teach the concepts, but it is not the only approach. When you flip a coin are you really taking a random sample of all the possible flips? When Fisher analyzed agricultural data over time he did not randomly sample from all the years that would ever grow plants. As long as there is a probability model that links the parameters to the data you can use statistical techniques. A permutation test seems appropriate to the original question to see if the difference can be attributed to chance. Commented Apr 27, 2012 at 19:20
• There's some discussion on this elsewhere on CV - stats.stackexchange.com/questions/2628/…. While you need to be careful what inferential approaches you use, I cannot accept there is no point in many of the standard tests which test a null hypothesis that the population can be adequately modelled as produced by a specified data generating process. There are many times we have the whole population but we want to model the data generating process - often much more interesting than the actual individuals who were generated. Commented Apr 28, 2012 at 3:26

@Macro should've posted his comment as an answer, but just to add a tad of rigor to this, here are some references:

Binder, D and Roberts, G (2003) Design-based and model-based methods for estimating model parameters

Brewer, K (2002) Combined Survey Sampling Inference

• re: macro, agreed. overall, I'm really grateful for the support from you guys! Commented May 4, 2012 at 3:52