I want to develop a multiple linear regression with 457 samples. Among independent variables, there is one categorical variable which consists of three level (XL1, XL2 AND XL3). The number of XL1, XL2 and XL3 are 229, 214 and 14 respectively. Looking at proportion levels, XL3 is only 0.030 which can be neglected or at least merged with other levels.
Is any statistical test which can be decide whether the number of level of a design variable could be reduced?.
Any way, I have reviewed one paper which stated the hypothesis as below:
H0 : pi = 0
H1: pi ≠ 0
where pi is the proportion of class i, with 0.05 significance level
The paper claimed that if pi < 0.05, then class i can be reduced from a design variable.
I wonder how it can be since pi value fall in alternative H1 which is pi ≠ 0. Is it correct approach? If yes, please provide your explanation.
You do not have to collapse groups unless:
1) You want to do it for some scientific reason (e.g. XL2 and XL3 are closely related)
2) You are fitting a model with many covariates, whereby having too many categories in your categorical variables will overfit your model.
When you include the three categories in your model, one of them, let's say XL1 will be the baseline category with which the others will be compared. The model will give you coefficients and p-values for XL2 and XL3 that will tell you if each of these is significantly associated with the outcome.
If you decide/need to collapse you will lose information, so I would suggest you avoid doing that.
In fact, your question has two rather different parts. Fr the first part, I concur with the previous answers: Do not reduce the number of levels since there are only three of them, at least if they stand for markedly different properties. However, it really depends on the context. Here is a counterexample: If, just as an example, the statistical unit is a car, the three levels stand for three different engine type 1=gas, 2=Diesel and 3=all-electric and you would like to determine whether all-electric cars or ones with conventional combustion engines are more attractive, then merge the levels "gas engine" and "Diesel engine" to "combustion engine" thereby giving slightly higher significance to all the statistical output.
Regarding you "pi" question, the reviewed paper you mentioned should clearly be rejected, at least on this account. If H0 states that the percentage pi of a certain attribute level x0 of a variable X is zero, then a single counterexample (e.g., data where this level occurs at least once) rejects H0 with absolute certainty (p value exactly equals to zero). Most generally, the p value is defined as the probability that, assuming H0, the outcome deviates from (the boundary of) H0 at least as strongly as the data. In this example, the number Y of occurring values x0 is binomial(n,theta) distributed with n the sample size and theta=0 according to H0. This leads to P(Y)=0 for any Y not equal zero (just look up the binomial distribution). So any outcome yData > 0 has a p value of zero, regardless of the percentage of this level. What this guy probably did was confusing the p value of 5% (typically, significance is assumed if p is less) with the (totally unrelated) percentage yData/n of levels x0 of a certain variable X
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