Regression Coefficients: Unstandardized Versus Standardized

Regression Coefficients: Unstandardized versus Standardized

Using the first 50 cases in the BodyFat data file, we employ linear regression to predict percentage of body fat from age and ankle circumference. Here are the regression coefficients.

Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / -21.173 / 11.967 / -1.769 / .083
Ankle_Inches / 2.233 / 1.229 / .220 / 1.818 / .075
Age_Years / .541 / .131 / .501 / 4.140 / .000
a. Dependent Variable: PercentFat1

Look first at the unstandardized slopes. Which indicates the stronger effect? The slope for ankle circumference is 2.233, the slope for age is .541, so ankle circumference is the larger effect, right? Wrong! Look at the standard deviations of the predictors:

Descriptive Statistics
N / Minimum / Maximum / Mean / Std. Deviation
PercentFat1 / 50 / 4.6 / 38.2 / 17.788 / 9.0813
Ankle_Inches / 50 / 8.11 / 13.35 / 9.2858 / .89431
Age_Years / 50 / 22 / 50 / 33.66 / 8.402
Valid N (listwise) / 50

A one inch increase in ankle circumference is a larger change (1/.89 = 1.12) than is a one year increase in age (1/8.4 = .12), so even if the effect of ankle circumference were exactly as strong as the effect of age, the unstandardized slope for ankle circumference would be greater. It is the standardized slopes that give us estimates of the strength of each predictor evaluated in identical metrics (standard deviation units), and the standardized slope for age (.50) is considerably larger than that of ankle circumference (.22). Also notice that the unstandardized slope for age, .541, is significantly greater than zero (p < .001), but the larger unstandardized slope for ankle circumference, 2.233, is not (p = .075).

Now suppose that with the same subjects I measured age in days rather than years. Here are the regression coefficients:

Now the effect of age appears to be miniscule, but that is just because an increase in age of just one day is a tiny change, so the accompanying increase in body fat will also be small. Notice that the standardized slopes remain exactly the same as they were before, as do the values of t and p.

Suppose I measured age in years and ankle circumference in mm. Here are the regression coefficients:

Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / -21.173 / 11.967 / -1.769 / .083
Ankle_mm / .088 / .048 / .220 / 1.818 / .075
Age_Years / .541 / .131 / .501 / 4.140 / .000

Now the effect of ankle circumference appears very small, but its standardized slope remains unchanged. A one mm change in ankle circumference is very small change, and accordingly its unstandardized slope is small.

When you are dealing with variables like age and ankle circumference, you may hope that your audience will take into account the units of measure when interpreting unstandardized slopes. With the variables psychologists typically use, there is no cause for such hope, as the units of measure are arbitrary. Suppose I told you that for each one point increase in my measure of misanthropy there was a 3.68 point increase in my measure of political conservatism. Is that a small slope, medium slope, or large slope? You will have no idea how large that slope is until you see it in standardized form.

So, should one report only the standardized slopes? Probably not. It would best to report both the unstandardized slopes and the standardized slopes. Having the unstandardized slopes makes it easier to compare the results of two studies that used the same variables but different subjects. Suppose that we repeated the research described above but with a population in which percent body fat was much less variable (SD = 4). Suppose that the unstandardized partial slope for age remained at .541 and the SD for age remained at 8.402. In both samples the mean change in percent body fat is .541 per one year increase in age. For the first sample the beta for age is . In the second sample it is . Does this difference in beta weights indicate that the effect of age on percent body fat is greater in the second sample than in the first or is it just an artifact of the reduced variance in percent body fat in the second sample?

What About Indirect Effects in a Mediation Analysis

Having convinced a graduate student that she should report both unstandardized and standardized coefficients in the mediation analyses reported in her dissertation, she asked “because I reported unstandardized coefficients, am I also unable to interpret the pairwise comparisons of individual indirect effects within a single model.” I shall answer her question with a hypothetical example.

I am researching the effect of a plant nutrient on production of a crop used as fuel to generate electricity. One group of plants gets the standard fertilizers, the other gets the same but also the nutrient being investigated. My model is that the experimental nutrient will cause the treated plants to grow a more extensive system of roots (measured in meters of depth), which will result in a greater yield (kilograms) which will result in generation of more electricity when the fuel is burned (kilowatts). A, B, C, D, & E are unstandardized coefficients. /

Process provides coefficients for the three indirect effects and pairwise comparisons between effects.

Indirect effect(s) of X on Y /
/ Effect / Boot SE / BootLLCI / BootULCI /
Total / ~ / ~ / ~ / ~
Ind1: / ~ / ~ / ~ / ~
Ind2: / ~ / ~ / ~ / ~
Ind3: / ~ / ~ / ~ / ~
(C1) / ~ / ~ / ~ / ~
(C2) / ~ / ~ / ~ / ~
(C3) / ~ / ~ / ~ / ~
Indirect effect key /
Ind1: / Group / -> / Meters / -> / Kilowatts
Ind2: / Group / -> / Meters / -> / Kgram / -> / Kilowatts
Ind3: / Group / -> / Kgram / -> / Kilowatts
Specific indirect effect
contrast definitions /
(C1) / Ind1 / minus / Ind2
(C2) / Ind1 / minus / Ind3
(C3) / Ind2 / minus / Ind3

Indirect effect 1 is the product of coefficients A and B. Here I show that product in terms of the units of measure: -- that is, the indirect coefficient tells you how many kilowatts energy production increases per one unit change in the code for groups (coded 0,1), that is, what is the difference between groups due to this indirect effect.

Indirect effect 2 is .

Indirect effect 3 is .

Notice that the unit of measure for each of the indirect effects is group difference in kilowatts of electricity produced. I could change the unit of measure for the mediators to feet and ounces, miles and tons, light years and stones, or whatever, the unit of measure for each indirect effect would remain kilowatts difference between groups for that indirect effect.

Ø Fair Use of This Document

Ø Return to Wuensch’s Stats Lessons Page