# Get Started

This article is a brief introduction to kcmeans.

library(kcmeans)
set.seed(51944)

To illustrate kcmeans, consider simulating a small dataset with a continuous outcome variable y, two observed predictors – a categorical variable Z and a continuous variable X – and an (unobserved) Gaussian error. As in Wiemann (2023), the reduced form has an unobserved lower-dimensional representation dependent on the latent categorical variable Z0.

# Sample parameters
nobs = 800 # sample size
# Sample data
X <- rnorm(nobs)
Z <- sample(1:20, nobs, replace = T)
Z0 <- Z %% 4 # lower-dimensional latent categorical variable
y <- Z0 + X + rnorm(nobs)

kcmeans is then computed by combining the categorical feature with the continuous feature. By default, the categorical feature is the first column. Alternatively, the column corresponding to the categorical feature can be set via the which_is_cat argument. Computation is very quick – indeed the dynamic programming algorithm of the leveraged Ckmeans.1d.dp package is polynomial in the number of values taken by the categorical feature Z. See also ?kcmeans for details.

system.time({
kcmeans_fit <- kcmeans(y = y, X = cbind(Z, X), K = 4)
})
##    user  system elapsed
##   0.784   0.027   0.668

We may now use the predict.kcmeans method to construct fitted values and/or compute predictions of the lower-dimensional latent categorical feature Z0. See also ?predict.kcmeans for details.

# Predicted values for the outcome + R^2
y_hat <- predict(kcmeans_fit, cbind(Z, X))
round(1 - mean((y - y_hat)^2) / mean((y - mean(y))^2), 3)
## [1] 0.695
# Predicted values for the latent categorical feature + missclassification rate
Z0_hat <- predict(kcmeans_fit, cbind(Z, X), clusters = T) - 1
mean((Z0 - Z0_hat)!=0)
## [1] 0

Finally, it is also straightforward to compute standard errors for the final coefficients, e.g., using summary.lm:

# Compute the linear regression object and call summary.lm
lm_fit <- lm(y ~ as.factor(Z0_hat) + X)
summary(lm_fit)
##
## Call:
## lm(formula = y ~ as.factor(Z0_hat) + X)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -3.1205 -0.6916  0.0544  0.6700  3.4201
##
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)         0.03897    0.07434   0.524      0.6
## as.factor(Z0_hat)1  0.88393    0.10265   8.611   <2e-16 ***
## as.factor(Z0_hat)2  1.88314    0.10271  18.334   <2e-16 ***
## as.factor(Z0_hat)3  3.01094    0.10636  28.310   <2e-16 ***
## X                   1.04636    0.03541  29.549   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.03 on 795 degrees of freedom
## Multiple R-squared:  0.6954, Adjusted R-squared:  0.6939
## F-statistic: 453.7 on 4 and 795 DF,  p-value: < 2.2e-16

# References

Wiemann T (2023). “Optimal Categorical Instruments.” https://arxiv.org/abs/2311.17021