In ekstroem/mommix: Moment-Based Estimation of Regression Mixtures

knitr::opts_chunk$set(
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Volatile acidity (VA) refers to the process when lactic acid bacteria and acetic acids turns wine into vinegar, and the process takes place mainly due to growth of bacteria, the oxidation of ethanol, or the metabolism of acids/sugars. Wines with a high level of pH are supposedly more susceptible to oxidation and the antibacterial effects of sulfur dioxide and of fumaric acid are reduced rapidly as the pH level increases. Consequently, wines are thought to lose their quality as they become less acidic (increased pH) since the volatile acidity increases.

The paper by Cortez et al (2009) considers 11 physicochemical properties of a selected sample of Portuguese vinho verde wines. Samples from 1599 red wines and 4898 white wines are available and the relationship between volatile acidity and pH are shown separately for red and white wines in the figure below.

library(mommix)
library(dplyr)
library(ggplot2)
data(wine)
## Remove extreme observations
wine2 <- wine %>% filter(residual.sugar<25)

## Plot the data
wine2 %>% ggplot(aes(x=pH, y=volatile.acidity, col=colour)) +
                geom_point(alpha = 1/5, position = position_jitter(h = 0), size = 2) +
        ylab("Volatile acidity") + 
                geom_smooth(method = 'lm', se=FALSE)

While it is apparent that the available red wines generally have slightly higher levels of pH, it also appears as if the impact of pH on volatile acidity is largest for the red wines in the sample: Individual regression lines for the two types of wine show an almost horizontal line for white wine while there is an effect of pH on VA for the red wines:

winelm2 <- lm(volatile.acidity ~ pH*colour-1, data=wine2)
summary(winelm2)

If we did not know that the full 6497 samples were comprised of two different types of wine we might pursue regressing volatile acidity on pH for the full data.

winelm <- lm(volatile.acidity ~ pH, data=wine2)
summary(winelm)

This gives the dashed regression shown in the figure below, which suggests an overall effect of pH on volatile acidity.

wine2 %>% ggplot(aes(x=pH, y=volatile.acidity, col=colour)) + geom_point(alpha = 1/5, position = position_jitter(h = 0), size = 2) +
  ylab("Volatile acidity") + 
  geom_smooth(method = 'lm', se=FALSE) + 
  geom_abline(intercept=coef(winelm)[1], slope=coef(winelm)[2], linetype=2) 

When we fit the moment-based mixture model then we see that the data are likely to consist of a mixture, (\hat{p} = 0.28) (95\% CI: 0.24-0.33), which suggests that only a proportion of the wines are influenced by pH:

wineres <- moment_mixture_model(wine2, volatile.acidity ~ pH, weight="square")
wineres

(For now we use the following code, which only handles one predictor but provides the standard errors. This will be removed in the future)

wineback <- cbpcode(wine2$volatile.acidity, wine2$pH)
wineback

Also, the effect driving the association between the volatile acidity and pH appears to be much stronger than what was observed from analyzing the full data with a simple linear regression model, (\hat\beta=) 0.94 (95\% CI 0.51-1.37). Consequently, the moment mixture model is able to identify that the wine data is likely to consist of two types of wines that respond differently to changes in levels of pH with only placing very few restrictions on the underlying distributions.

ekstroem/mommix documentation built on May 14, 2019, 9:36 p.m.