Menna Arafat

15 Correlation analysis

Correlation measures the strength and direction of association between two continuous variables.

Method Measures Useful when
Pearson linear association the relationship is approximately linear without influential outliers
Spearman monotonic rank association data are skewed, contain outliers, or the relationship is not strictly linear

15.1 Pearson correlation

Pearson’s correlation coefficient measures the strength and direction of a linear relationship:

\[ r=\frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum_i(x_i-\bar{x})^2\sum_i(y_i-\bar{y})^2}}. \]

It ranges from \(-1\) to \(1\). Values closer to \(-1\) or \(1\) indicate stronger linear association.

cor.test(~biomarker_baseline_ng_mL+biomarker_week12_ng_mL,
         data=dat,method="pearson")
#> 
#>  Pearson's product-moment correlation
#> 
#> data:  biomarker_baseline_ng_mL and biomarker_week12_ng_mL
#> t = 52, df = 347, p-value <0.0000000000000002
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.9286 0.9526
#> sample estimates:
#>    cor 
#> 0.9418

A scatterplot should accompany Pearson correlation to assess the form of the relationship and identify influential observations.

ggplot(dat,aes(biomarker_baseline_ng_mL,biomarker_week12_ng_mL,colour=treatment_arm)) +
  geom_point(alpha=.45,na.rm=TRUE) +
  geom_smooth(method="lm",se=FALSE,na.rm=TRUE) +
  scale_colour_manual(values=pal) +
  labs(x="Baseline biomarker (ng/mL)",y="Week-12 biomarker (ng/mL)",colour="Treatment")
Baseline and week-12 biomarker with a linear fit.

Figure 15.1: Baseline and week-12 biomarker with a linear fit.

15.2 Spearman rank correlation

Spearman correlation uses the ranks of the observations rather than their original values. It measures the strength and direction of a monotonic relationship and is useful for skewed data.

Because CRP is strongly right-skewed, Spearman correlation is appropriate for describing the association between baseline and week-12 CRP.

cor.test(~crp_baseline_mg_L+crp_week12_mg_L,
         data=dat,method="spearman",exact=FALSE)
#> 
#>  Spearman's rank correlation rho
#> 
#> data:  crp_baseline_mg_L and crp_week12_mg_L
#> S = 264568, p-value <0.0000000000000002
#> alternative hypothesis: true rho is not equal to 0
#> sample estimates:
#>    rho 
#> 0.9582

A strong Pearson or Spearman correlation does not imply that two measurements agree, nor does it establish causation.

15.3 Correlation matrix

A correlation matrix summarizes associations among several continuous variables. Here, Spearman correlations are used because some variables are skewed.

cor_vars <- dat |> transmute(
  Age=age_years,BMI=bmi_kg_m2,`Baseline SBP`=sbp_baseline_mmHg,
  HbA1c=hba1c_pct,LDL=ldl_mg_dL,HDL=hdl_mg_dL,eGFR=egfr_mL_min_1_73m2,
  `Baseline CRP`=crp_baseline_mg_L,
  `Baseline biomarker`=biomarker_baseline_ng_mL,
  `Baseline QoL`=qol_baseline_0_100)

cm <- cor(cor_vars,use="pairwise.complete.obs",method="spearman")
round(cm,2)
#>                      Age   BMI Baseline SBP HbA1c   LDL   HDL  eGFR Baseline CRP Baseline biomarker Baseline QoL
#> Age                 1.00  0.06         0.35  0.24  0.03 -0.02 -0.76         0.15               0.13        -0.32
#> BMI                 0.06  1.00         0.28  0.30  0.01 -0.10 -0.04         0.04               0.12        -0.21
#> Baseline SBP        0.35  0.28         1.00  0.20  0.08 -0.05 -0.21         0.02               0.13        -0.20
#> HbA1c               0.24  0.30         0.20  1.00  0.07  0.03 -0.21         0.00               0.09        -0.13
#> LDL                 0.03  0.01         0.08  0.07  1.00 -0.01  0.01        -0.09              -0.03        -0.01
#> HDL                -0.02 -0.10        -0.05  0.03 -0.01  1.00  0.00        -0.06              -0.04         0.10
#> eGFR               -0.76 -0.04        -0.21 -0.21  0.01  0.00  1.00        -0.11              -0.07         0.26
#> Baseline CRP        0.15  0.04         0.02  0.00 -0.09 -0.06 -0.11         1.00               0.07        -0.19
#> Baseline biomarker  0.13  0.12         0.13  0.09 -0.03 -0.04 -0.07         0.07               1.00        -0.14
#> Baseline QoL       -0.32 -0.21        -0.20 -0.13 -0.01  0.10  0.26        -0.19              -0.14         1.00
cm_long <- as.data.frame(cm) |>
  tibble::rownames_to_column("x") |>
  pivot_longer(-x,names_to="y",values_to="rho")

ggplot(cm_long,aes(x,y,fill=rho)) +
  geom_tile(colour="white",linewidth=.4) +
  geom_text(aes(label=sprintf("%.2f",rho)),size=3) +
  scale_fill_gradient2(low="#B45252",mid="white",high="#3F7F8D",limits=c(-1,1)) +
  coord_equal() +
  labs(x=NULL,y=NULL,fill="Spearman ρ") +
  theme(axis.text.x=element_text(angle=45,hjust=1))
Spearman correlation matrix for selected baseline variables.

Figure 15.2: Spearman correlation matrix for selected baseline variables.

15.4 Correlation is not causation

Correlation shows whether two variables move together, but it does not show that one causes the other.

A strong correlation between baseline and follow-up measurements may indicate that baseline is useful to include in a regression model. However, the treatment effect is estimated by the regression model, not by the correlation coefficient.

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