# Significance in simple regression but not multiple regression

Background: I am trying to show an increase in a number of variables over time, so initially I ran a simple linear regression. However, I subsetted the data so that I could get a regression estimate for each species and reef combination (which would indicate the change per year).

The entire regression would look something like

lm(Density ~ Time, data=data, subset=ReefSpecies=="ReefSpecies1).

Below I show the output for Density one of the response variables, where Little Grecian, etc = Reef and O. annularis, etc = Species. There are 3 significant and 2 marginally significant ReefSpecies with time. The second column is the estimate, the third column is the F-value, and the last column is the p-value.

Of course, I was happy with the output but I also wanted to make a linear model to see what was accounting for this increase/decrease/no change in my response variables.

The linear model looks something like this: lm(Density ~ Species + I(Chl_zoox^0.25) + Season + TN + Biomass + Turbidity + FvFm + Time, data=Seasonal) and the output indicates that time is nonsignificant.

Last, I was told that if I ran a regression with the a 3-way interaction of Reef*Species*Time (e.g. lm(Density ~ I(Chl_zoox^0.25) + Season + TN + Biomass + Turbidity + FvFm + Species*Reef*Time, data=Seasonal)) that I would get theoretically get the same estimates as the simple linear regression (lm(Density ~ Time, data=data, subset=ReefSpecies=="ReefSpecies1)). Clearly, this is not the case, and I receive NA for some of the output. Question:

1. Why is there a difference between the significance in the multiple regression and the simple regression?

From my understanding, the multiple regression has more power and therefore should still be significant.

2. I think I might be interpreting the interactions incorrectly but it seems the estimates in the simple regression do not equal (but are close) the multiple regression estimates (e.g - Jaap - O. faveolata = -0.2504 in the simple regression but in the multiple regression 0.001487 + 0.092568 + 0.559377 - 0.818717 = -0.1652).

How can I interpret this interaction to get similar values? And why wouldn't they be significant?

3. What is the deal with NA in the interaction output? Does this have to do with multicollinearity?

Third, your initial data summary shows that there are no O.franksi in any reef environments except for Alligator. So there is no way to obtain coefficients that include interactions of O.franksi with Reef; you have no data on 3 of the 4 Reef environments. Hence the NA values.