1. Multivariate Analyses: Correlation
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Multivariate Analyses: Correlation
Linear regression: independent variable has some influence on dependent variable.
Correlation: two or more dependent variables reasonably assumed to be (partially) controlled by same independent variable(s).

2. Multivariate Analyses: Correlation
Correlation: why can’t we just do LSLR?
Is it reasonable that one variable causes/influences variation in another?
Is it reasonable that both (or more) variables are influenced by same independent variables?
Are neither of these propositions reasonable?
Can we identify a (independent) variable that is fixed, known without error?

3. Pearson’s Correlation Coefficient
Pearson’s Product-Moment Correlation (Pearson’s correlation)
Appears to be a linear or continuous distribution of points
No extreme outliers, clusters of points, or curvilinear relationship
𝑟 1,2 = 𝑦 1 𝑦 𝑦 𝑦 2 2
Pearson’s r can be squared and = coefficient of determination (r2) for same data
= 0.41

4. Pearson’s Correlation Coefficient
Pearson’s Product-Moment Correlation (Pearson’s correlation)
𝑟 1,2 = 𝑦 1 𝑦 𝑦 𝑦 2 2
What is a significant correlation?
H0: ρ = 0
t = 𝑟 −0 𝑠 𝑟
Critical value for t[.05, n-2]
> cor(howdata$GOL, howdata$FOL)
[1] 0.408
> cor.test(howdata$GOL, howdata$FOL)
Pearson's product-moment correlation
data: howdata$GOL and howdata$FOL
t = 20, df = 2000, p-value <2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
sample estimates:
cor
0.408
1− 𝑟 2 𝑛 −2

5. Pearson’s Correlation Coefficient
Pearson’s Product-Moment Correlation (Pearson’s correlation)
Assumptions:
Variables are not causal related
Variables are not mechanically linked so that they are not independent
Variables are normally distributed.
In anthropology, variables are often not normally distributed, so we use non-parametric methods

6. Spearman’s Rank-Order Correlation coefficienty1
rank1 y2
rank2
189 1 36
2.5
183 3
174
6.5 35 5
179 37 32 7
181 4 34 6
172 33
188 2
168 9 31 8
170
Perfect positive rank order correlation: highest rank for one variable correlated with highest rank for the other
Perfect negative rank order correlation: highest rank of one variable correlated with lowest rank for the other

7. Spearman’s Rank-Order Correlation coefficient
Are the rank order abundances of fish species correlated at different sites
What is the independent variable here?
Compare Spearman’s coefficient to table of critical values at alpha and for sample size = number of variate comparisons (10 here)
> cor.test(fish$site1, fish$site2, method = "spearman“, exact = FALSE)
Spearman's rank correlation rho
data: fish$site1 and fish$site2
S = 40, p-value = 0.007
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.784

8. Correlation Summary Regression: dependent and independent variables
Correlation: 2+ dependent variables
Rank order correlation: typically assumptions of normality not met
Consider what you question you are trying to answer.