# How to get 95% CIs for standardized regression coefficients?

I am running multiple linear regression with categorical variables and I need confidence interval 95% for standardized regression coefficient. I searched around and found 2 methods:

1. Using the QuantPsyc package, with the function lm.beta. However, using lm.beta I can only get the standardized coefficients whereas I need with their 95% CI too. Is there a way?

2. To extract standardized regression coefficient, first standardize all the variables involved, and then run it in linear regression then you'll get estimates for standardized coefficients.

So here is my model:

model1 <- lm(Life_Satisfaction ~ Subjective + Age + Sex + CountryCat11 +
CountryCat12 + CountryCat13 + CountryCat14 +
CountryCat15 + CountryCat16 + CountryCat17 +
CountryCat18 + CountryCat19 + CountryCat20 +
CountryCat23 + CountryCat25 + CountryCat28 +
CountryCat29 + CountryCat30 + Education_ISCED1 +
Education_ISCED2 + Education_ISCED3 +
Education_ISCED4 + Education_ISCED5 +
Education_ISCED6 + Education_stillinschool +
Education_None + Education_other, data=lifesat)

lm.beta (model1)


I ran that, but I cannot get the 95% CI.

So I tried the scale method:

model2 <- lm(scale(Life_Satisfaction) ~ scale(Subjective) + scale(Age) +
scale(Sex) + scale(CountryCat11) +
scale(CountryCat12) + scale(CountryCat13) +
scale(CountryCat14) + scale(CountryCat15) +
scale(CountryCat16) + scale(CountryCat17) +
scale(CountryCat18) + scale(CountryCat19) +
scale(CountryCat20) + scale(CountryCat23) +
scale(CountryCat25) + scale(CountryCat28) +
scale(CountryCat29) + scale(CountryCat30) +
scale(Education_ISCED1) + scale(Education_ISCED2) +
scale(Education_ISCED3) + scale(Education_ISCED4) +
scale(Education_ISCED5) + scale(Education_ISCED6) +
scale(Education_stillinschool) + scale(Education_None) +
scale(Education_other), data=lifesat)

summary(model2)


I ran that, and I got the standardized regression and 95% CI but it was different from the standardized regression results I got from SPSS? Did I do it wrong?

• There is enough statistical confusion here that this could be considered on topic, IMO. – gung - Reinstate Monica Sep 9 '15 at 23:37
• Questions solely about how software (eg R) works are off topic here, but you may have a real statistical question buried here. You may want to edit your question to clarify the underlying statistical issue. You may find that when you understand the statistical concepts involved, the software-specific elements are self-evident or at least easy to get from the documentation. – gung - Reinstate Monica Sep 9 '15 at 23:38

Maybe you did not standardized the outcome. If you just standardized the covariates then the results will be a little bit different from the results of the standardized beta in SPSS for which both the outcome and covariates are standardized.

For simplicity, assume that there is one focal continuous predictor $$x$$ and a continous outcome $$y$$. Standardization doesn't really make a lot of sense with categorical predictors, imo. The regression model could include more predictors but the following answer focuses only on one of them. Then, we have four possibilities:

1. Both $$y$$ and $$x$$ are standardized (meaning both have mean $$0$$ and standard deviation $$1$$). Denote the regression coefficient of $$x$$ as $$\beta_{xy}$$.
2. Only $$x$$ is standardized. Denote the regression coefficient of $$x$$ as $$\beta_{x}$$.
3. Only $$y$$ is standardized. Denote the regression coefficient of $$x$$ as $$\beta_{y}$$.
4. Neither $$y$$ or $$x$$ are standardized. Denote the regression coefficient of $$x$$ as $$\beta$$.

Further, let $$s_x$$ and $$s_y$$ be the standard deviations of $$x$$ and $$y$$, respectively.

In the following section, I'm going to show how to convert the regression coefficients from the standardized models (cases 1-3) to the coefficient in the unstandarized model (case 4) and vice versa. The crucial thing to note is that the same conversion formulas can be applied for converting standard errors and/or confidence limits! An illustration of case 1 in R is at the bottom of this answer.

Case 1: Both $$y$$ and $$x$$ are standardized

To convert from $$\beta$$ to $$\beta_{xy}$$ without running another model: $$\beta_{xy} = \beta\cdot \frac{s_x}{s_y}$$.
To convert from $$\beta_{xy}$$ to $$\beta$$ without running another model: $$\beta = \beta_{xy}\cdot \frac{s_y}{s_x}$$.

To answer your first question: Calculate the regression model with no standardized variables. Multiply the confidence limits for the regression coefficients with $$\frac{s_x}{s_y}$$.

Case 2: Only $$x$$ is standardized

To convert from $$\beta$$ to $$\beta_{x}$$ without running another model: $$\beta_{x} = \beta\cdot s_x$$.
To convert from $$\beta_{x}$$ to $$\beta$$ without running another model: $$\beta = \beta_{x}\cdot \frac{1}{s_x}$$.

Case 3: Only $$y$$ is standardized

To convert from $$\beta$$ to $$\beta_{y}$$ without running another model: $$\beta_{y} = \beta\cdot \frac{1}{s_y}$$.
To convert from $$\beta_{y}$$ to $$\beta$$ without running another model: $$\beta = \beta_{y}\cdot s_y$$.

Here is a short illustration in R for the first case. The focal predictor is Fertility:

# Standard deviations
sx <- sd(swiss$$Fertility) sy <- sd(swiss$$Infant.Mortality)

# Models
mod_unstand <- lm(Infant.Mortality~Fertility + Agriculture, data = swiss)
mod_fully_stand <- lm(scale(Infant.Mortality)~scale(Fertility) + scale(Agriculture), data = swiss)

coef(mod_unstand)[2]

Fertility
0.1166856

# Convert unstandardized coefficient of "Fertility" to a fully standardized one
0.11668557*(sx/sy)

[1] 0.50043

# Check
coef(mod_fully_stand)[2]

scale(Fertility)
0.50043


For the confidence intervals, we use the same conversions:

# Confidence interval for the unstandardized coefficient
confint(mod_unstand)[2, ]

2.5 %     97.5 %
0.04993591 0.18343524

# Convert the confidence limits from the unstandardized model to a full standardized model
confint(mod_unstand)[2, ]*(sx/sy)

2.5 %    97.5 %
0.2141604 0.7866996

# Check
confint(mod_stand)[2, ]

2.5 %    97.5 %
0.2141604 0.7866996