Associations - Regression and Correlation

The regression model

  • The ordinary regression model \(y_i = \beta_0 + \beta x_i + \epsilon_i\) gives us the relationship between two variables, \(x\) and \(y\).
  • The correlation can be used to describe the same relationship using a unit-less number between -1 and 1.
  • A correlation of 0 indicate no association, a correlation close to 1 and -1 indicate association.

The relationship between training volume and vastus lateralis thickness

Are assumptions met?

  • No apparent curve-linear relationship
  • There are no obvious outliers, and
  • Both variables are evenly distributed (normally distributed)

The relationship between training volume and vastus lateralis thickness

library(exscidata)

cor(hypertrophy$VL_T1, 
    hypertrophy$SQUAT_VOLUME, 
    use = "complete.obs")

Using the correlation for inference

  • We might want to say something about the population from where we have gathered participants
  • The correlation coefficient can be used for hypothesis testing using the cor.test function
cor.test(hypertrophy$VL_T1, 
         hypertrophy$SQUAT_VOLUME, 
    na.action = na.omit)

    Pearson's product-moment correlation

data:  hypertrophy$VL_T1 and hypertrophy$SQUAT_VOLUME
t = 3.335, df = 27, p-value = 0.00249
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.2164913 0.7568216
sample estimates:
      cor 
0.5401395

Correlation and univariate regression

  • The correlation coefficient is a simplification of a univariate regression.
  • We will get the same results from lm (\(\sqrt{R^2}\)) as the estimate from cor.test
reg_mod <- lm(VL_T1 ~ SQUAT_VOLUME, 
              data = hypertrophy)
        
cor_res <- with(hypertrophy, 
                cor.test(VL_T1, 
                         SQUAT_VOLUME))      

sqrt(summary(reg_mod)$r.squared)

cor_res$estimate
R from lm 
0.5401395 
Correlation estimate from cor.test 
                         0.5401395

The correlation comes in many forms

  • When assumptions about normally distributed variables and outliers are questioned we may use a rank-based correlation.

  • Spearman’s or Kendall’s correlation coefficient are both alternatives to the Pearson correlation coefficient

  • Spearman’s \(\rho\) is simply the correlation coefficient calculated from ranked values

In R

## Ranking is sensitive to missing values
dat <- hypertrophy %>%
        select(VL_T1, SQUAT_VOLUME) %>%
         filter(!is.na(VL_T1), 
               !is.na(SQUAT_VOLUME))

# The spearman method
with(dat, 
     cor.test(VL_T1, 
              SQUAT_VOLUME, 
              method = "spearman"))

# Pearson method on ranked data
with(dat, 
     cor.test(rank(VL_T1), 
              rank(SQUAT_VOLUME), 
              method = "pearson"))

    Spearman's rank correlation rho

data:  VL_T1 and SQUAT_VOLUME
S = 1609.8, p-value = 0.0005285
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.6035045 

    Pearson's product-moment correlation

data:  rank(VL_T1) and rank(SQUAT_VOLUME)
t = 3.9329, df = 27, p-value = 0.0005285
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.3043080 0.7943169
sample estimates:
      cor 
0.6035045

Many types of associations

  • A broader definition of association can include…
    • Diffrences in body weight between sexes → sex (discrete) is associated with body weight (continuous)
    • Prevalence of ACL injuries differ between sexes → sex (discrete) is associated with injury status (discrete).
  • We use numerous statistical tests to describe associations (t-test, ANOVA, regression, etc.)

Associations and causality

Figure 1: US spending on science, space and technology is positively related to the number of suicied by hanging, strangulation and suffocation.

Associations and causality

  • Associations are descriptive → We observe statistical associations without being able to infer causality
  • Controlled experiments can be used to infer causality → Using experiments we can intervene in a system and interpret associations as causal (intervention associated with outcome)
  • Using graphs we can draw assumptions about relationships among variables

The experimental treatment (E) is causally associated with the outcome (O)

Summary

  • The correlation analysis can be seen as a simplified regression analysis
  • Correlation and regression are used to quantify associations
  • Associations are what we describe with (many) statistical tests
  • Association do not imply causation
  • Graphs can be used to draw assumptions about associations