# How to interpret sum of z-scores and average sum of z-scores?

Suppose I have data from 5 sub-tests and 2 groups of participants. What is the difference in the interpretation of a sum of z-scores (i.e, transform all 5 measures into z-scores and sum them up) versus the average sum (i.e, dividing the sum by 5) ?

• For example, If I wanted to analyse a model such as lm(scores ~ age + group). What does a "1-unit change" in the predictor variable mean in each context (sum x average sum) ?

Note. In my actual data, if I had enough participants, I would have run a PCA to create the composite score, but I can't do that, this is why I'll go with the z-scores. I need a composite score of the subtests.
Note2: These are accuracy scores (number of correct answers).

• Some dummy-data:
library(tidyverse)
set.seed(123)

# Generate dummy data
n <- 100  # Number of participants
data <- data.frame(
participant_id = 1:n,
age = sample(18:30, n, replace = TRUE),  # Age of participants
group = sample(c("male", "female"), n, replace = TRUE),  # Group
subtest1 = rnorm(n), subtest2 = rnorm(n), subtest3 = rnorm(n),
subtest4 = rnorm(n), subtest5 = rnorm(n)
)

# Standardize the subtest scores
data_z <- data %>%
mutate(across(starts_with("subtest"), scale))  %>% # Scale
(standardize) subtest scores
mutate(sum_z_scores = rowSums(select(.,starts_with("subtest"))),
# Sum of z-scores
avg_z_scores = rowMeans(select(., starts_with("subtest"))))
# Average of z-scores

# Fit linear models
model_sum <- lm(sum_z_scores ~ age + group, data = data_z)
model_avg <- lm(avg_z_scores ~ age + group, data = data_z)


The difference is a factor of 5 in the coefficients. You can interpret this as a unit change. It would be the same if your dependent variable was a distance and you changed units from meters to kilometers.

If you divide the left-hand side of an equation by a factor, you also need to divide the right-hand side by the same factor. Since the predictors are unchanged, the factor goes into the coefficients. The position of the sum of squares minimum is not impacted by such a unit change of the dependent variable.

$$y_i = \beta_0 + \beta_1x_{1,i} + \beta_2x_{2,i} + \varepsilon_i\\ \frac{y_i}{5} = \frac{\beta_0}{5} + \frac{\beta_1}{5}x_{1,i} + \frac{\beta_2}{5}x_{2,i} + \frac{\varepsilon_i}{5}$$

Hence:

coef(model_sum)
#(Intercept)          age    groupmale
#0.113514239 -0.001736993 -0.166028258
coef(model_avg) * 5
#(Intercept)          age    groupmale
#0.113514239 -0.001736993 -0.166028258

sum(model_sum$$residuals^2) #[1] 496.5761 sum(model_avg$$residuals^2) * 5^2
#[1] 496.5761


So, estimated effect sizes are stronger by a factor of five if you use summed scores compared to averaged scores. But they also have a different unit, you are just not seeing that unit. The actual effects are identical.

However, I don't understand why you sum or average the data. Why don't you do the regression with non-aggregated data? I.e., transform the data to long format and include the subtest as a predictor.

DF <- data %>%
mutate(across(starts_with("subtest"), scale)) %>%
pivot_longer(cols = starts_with("subtest"), names_to = "subtest", values_to = "score")

model <- lm(score ~ (age + group) * subtest, data = DF,
contrasts = list(group = contr.treatment, subtest = contr.sum))
#       (Intercept)                age          groupmale           subtest1           subtest2           subtest3           subtest4       age:subtest1
#      0.0227028479      -0.0003473987      -0.0332056516      -0.3876872943       0.4004517246       0.5819525806       0.4606010167       0.0154445180
#      age:subtest2       age:subtest3       age:subtest4 groupmale:subtest1 groupmale:subtest2 groupmale:subtest3 groupmale:subtest4
#     -0.0096406797      -0.0243325126      -0.0220695291       0.0305993804      -0.3875935877       0.0188624440       0.1734536370


Note how this gives you the same coefficient estimates as model_avg. But it tells you also something about how the age and group effects interact with the subtests. Another option could be fitting a mixed-effects model with random intercepts and slopes grouped by subtest.

• thank you. I still don't get it. But they also have a different unit, you are just not seeing that unit . I couldn't get my head around it. How do I interpret "a 1-unit change in age adds 0.11 to the summed composite test score (or .0275 to the avg score)". In what units would I be talking about? I edited the post for clarification as well. Commented Jul 29 at 9:54
• @LarissaCury The units are "score / test" for the average and "score / 5 tests" for the sum. Commented Jul 29 at 10:21
• Can I say, for the summed scores as DP: in relation to fem, males decrease -0.16 in the total accuracy score across all 5 tests and, for avg: in relation to fem, males decrease .0275 in the average accuracy score per test ? Are these correct interpretations? Commented Jul 29 at 10:52
• Yes, but I would say "males scored lower by ..." instead of "males decrease" and make it clear that those are marginal effects. Commented Jul 29 at 10:56
• And if the summed score is important for the research question, I would sum first and scale second. I believe that's what is commonly done for what is called "grading on a curve". Commented Jul 30 at 4:57