We have collected population growth data of a carnivore from five provinces and would like build a multiple linear regression model (OLS) with two explanatory variables. We have 7 observations each from two provinces, 22 observations each from two provinces, and 30 observations from one province, with the total sample size of 88. Data exploration shows a temporal trend in the growth rate so we need to include a time index in the model as a predictor to control its effect. We would like to explore the differences in growth rates across five provinces by including four dummies in the multiple regression model. But we are not sure whether that would really work because we have 7 observations in two states and there would be at least three predictors. Although we have enough samples at the aggregate level, we suspect that the sample size might be a limitation to do the analysis at the province level. We would highly appreciate your thoughts regarding the inclusion of four dummies to represent five provinces in the multiple regression model and how this will influence the estimates. We would greatly appreciate if you could direct us to the relevant literature on this issue.
Thank you for your suggestions.
Happy New Year!
Sincerely,
Nabin
Hi,
If you were only interested in comparing provinces, then you could view this as a problem of ANOVA with unequal N, straightforward to handle with OLS, but with relatively low power to detect differences from provinces with few (7) observations. If you used dummy coding (or in SPSS, indicator coding), that's three dummy variables, which saves you one degree of freedom compared to the four dummy variables you suppose in your question.
There is a very rough rule of thumb that ten observations per group is the minimum needed to adequately power an ANOVA, but the truth is that it depends on the standard errors of the means for each group and the size of the effect in the populations. Very precise estimates and very big effects don't need very large sample sizes to detect.
However, the problem is more complex than that since you say you are interested in comparing the growth rate between provinces. So not only are you comparing average growth rate between provinces (main effects in ANOVA terms) but also the slope of growth between provinces (province x time interactions in ANOVA terms). This will cost three more degrees of freedom. So three dummy variables for provinces, and three more for province x time interactions, plus one time effect is seven degrees of freedom for just the hypothesis you propose in your question.
On top of that, as you will be estimating slope in each province, in two of them only 7 observations provide information for the estimates, which isn't much. I would worry that your small sample size in two of the provinces would limit how much information you could learn about their differences with the other provinces.
Hope this helps,
Scot
Thank you Scot for answering the question in such a lucid way. Things are becoming clearer. I wonder whether you could suggest me some references from top of your head which deal with similar problems. Once again I greatly appreciate your time and effort for the answer.
In statistical scientific analysis, for each group it should selected min. 30 survey and totaly sholud be min. 30 survey. Because since Z dispersion (normal disp. theorem ) need min. 30 data. Otherwise if data are under 30 survey, results are not normal and scientific.
Hi,
Gelman and Hill's "Data Analysis Using Regression and Multilevel/Hierarchical Models" is what I think of first.
https://g.co/kgs/VtJiZC
and
http://www.stat.columbia.edu/~gelman/arm/
It assumes a working knowledge of R, so that can be a tough climb. But it does a nice job of describing simulation for sample size estimation and dealing with nested data like you have.
I hope that helps,
Scot
Thank you Hasan for the answer. Would you please let me know some references that talk about the requirement of at least 30 samples in each group when we are doing multiple regression with dummy variables.
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