Interpretation of percentage explanatory variable and scaling of beta

[vikas mishra]

I am running a regression model mentioned below.


HEI = beat1* fastfood_percentage + beta2* full dining_perentage + beta3* price_fast food* fastfood_percentage + ......... + error

beta1= -0.089
beta3 = -0.003
average price of meal from ff = $4.5


I have Healthy eating Index (HEI; range from 0 to 100) as dependent variable. percentage of calories from fast food(ff) , home and full service-dining(fd) are three percentage explanatory variables. These three sum up to 100%. So, I drop home calories as reference category while doing regression.

The interpretation would be 1 percentage point increase in ff calories cause a decline in HEI by 0.089 points taking home calories as reference and keeping fd calories constant.

Now if I want to interpret the effect for a meal ( assuming 33% of total calories consumed in a day) on HEI.

Queries :
(1) Can I simply scale the beta ( 33* -0.089= 2.93)?

(2) and interpretation would be an 33% percentage point increase in ff calories( substituting 33% of home calories, keeping fd percentage constant) cause a decline in HEI by 2.93 point.

(3) the interaction term would be interpreted as: In those who substitute home calories with fast food, a dollar increase in price and 1 percentage point substitution of home calories by fast food, cause a change in HEI by 0.003 points.


Please guide.

[Ken Chui]

I have Healthy eating Index (HEI; range from 0 to 100) as dependent variable. percentage of calories from fast food(ff) , home and full service-dining(fd) are three percentage explanatory variables. These three sum up to 100%.

I believe you missed a variable here. HEI + ff + fd should not amount to 100%. Did you miss something like "fh", as food at home?

The interpretation would be 1 percentage point increase in ff calories cause a decline in HEI by 0.089 points taking home calories as reference and keeping fd calories constant.

Not the case here, because there is an interaction term "beta3* price_fast food* fastfood_percentage". ff participates as itself and also part of an interaction term with fast food price. The above interpretation is only true when fast food price = 0, which is nearly always impossible.

Now if I want to interpret the effect for a meal ( assuming 33% of total calories consumed in a day) on HEI.

You'd need to be specific about what meal, ff, fd, and the missing home calories are all "meals".

Maybe try carefully revising and proofreading the question, and post again.

[vikas mishra]

Thanks Ken Chui for reply. Sorry, my post was not explaining the meaning of one meal and was not accounting the interaction. The revised post clarifies it :


Model:

HEI = constant+β1×ff_per+β2×fd_per+β3×p_meal_ff+β4×p _meal_ff×ff_per+...+error

Question: how substituting one meal from home food with one meal from fast food affects diet quality.

Where,

• One meal refers to 700Kcal and price of a one meal refers to price for 700Kcal.
• Average price of one meal from ff = $4.5 (for 700 Kcal or 1 meal)

Dependent variable: HEI; range from 0 to 100.
Independent variable:

1. Percentage of calories from fast food source(ff_per)
2. Percentage of calories from full-service dining source (fd_per)
3. Percentage of calories from home source (fh_per)
4. Price of one meal from ff: p_meal_ff

Restriction:
Further ff_per+ fd_per+fh_per= 100%

While regressing, I dropped fh_per variable as it will cause multicollinearity.

Beta estimate :
β1= -0.089
β4= -0.003

Interpretation:

The interpretation would be 1 percentage point increase in ff_per variale cause 0.10 HEI points decline.
{-0.089 + (-0.003) *4.5 = -0.1025} (marginal effect at mean (p_meal_ff))
Hence if we substitute one meal of home food(reference) with one meal from fast food keeping percentage of calories from full dining constant, the HEI drop by 3.3 (33*-0.10=-3.3) points.


Query:

To estimate the effect of one home meal substitution with one meal from ff, Can I simply scale the marginal effect ( 33* -0.10= -3.3)?