I am using categorical variable (having three categories) as independent variable in model and found that one category is coming to be significant while another category is not coming to be significant while variable is significant at overall level to be included in model. I am not able to understand whether I should include the insignificant category in the model.
Categorical variables can be represented several different ways in a regression model. The most common, by far, is reference cell coding. From your description (and my prior), I suspect that is what was used in your case. The standard statistical output will give you two tests. Let's say that A is the reference level, you will have a test of B vs. A, and a test of C vs. A (n.b., C can significantly differ from B, but not A, and not show up in these tests). These tests are usually not what you really want to know. You should test a multi-category variable by dropping both dummy variables and performing a nested model test. Unless you had an a-priori plan to test if a pre-specified level is necessary and it is not 'significant', you should retain the entire variable (i.e., all levels). If you did have such an a-priori hypothesis (i.e., that was the point of your study), you can drop only the level in question and perform a nested model test.
It may help you to read about some of these topics. Here are some references for further study:
Coding strategies for categorical variables:
- UCLA's stats help website
- I discuss reference cell coding here: Regression based for example on days of week
Problems with modifying your model based on what you find, when you didn't have a pre-specified hypothesis:
- While it's not framed exactly like your situation, you may be able to get the idea from my answer here: Algorithms for automatic model selection
Issues with multiple comparisons:
- You might skim some of the CV threads categorized under the multiple-comparisons tag
- the Wikipedia page for multiple comparisons
Nested model tests:
- Although discussed in terms of testing for moderation, my answer here should be clear enough to get the idea: Testing for moderation with continuous vs. categorical moderators
There is no need to include indicator variables for each of the categories. Let's say category A is coming out significant. Your results are suggesting that you consider collapsing the categories into "category A" and "all other categories".
Of course, you should perform an F-test for nested model vs. full model to check if removing indicator variables for other categories make sense.