Module 9 Correlation

Correlated variables

“move” together. If one variable increases, the other either tends to increase or decrease.

X increases as Y increases

 the variables have a positive correlation

Y decreases as X increases

the variables have a negative correlation

“r”

• The length or severity of correlation.

• linear correlation coefficient

r = 1

perfect positive correlation

r= -1

means perfect negative correlation

r= 0

 means no correlation 

|r| > 0.9

correlation is strong 

0.6 < |r| < 0.9

correlation is moderate

0.6 > |r|

the correlation is weak

Null hypothesis

Null hypothesis states that variables are not correlated.

• Inference for correlation

Reject  ρ = 0

results are large enough to be unlikely to occur by chance, assuming that the null hypothesis is tru

Statistically significant

 Often used when a null hypothesis is rejected

• Does not indicate strength of correlation

 ρ = 0 is true

there is no tendency for Y to change as X changes

Theoretical regression model

y_i = 𝛽₀+𝛽₁𝘹_i+𝟄_i

• Describes where data comes from

• Contains unknown parameters (𝛽₀,𝛽₁) and a random variation

y_i

The response or dependent variable 

𝛽₀

Population intercept

𝛽₁

population slope

𝘹_i

Predictor or independent variable

𝟄_i

 random variation around the line

Estimated model

 The equations for the line of best fit 

ŷ_i=b₀+b₁x_i

• Is the line itself and is drawn through the data

  • Contains statistical estimates with no random variation

Interpreting b₀

predicted value of y when x=0

Interpreting b₁

The slope, prediction for y for a one unit increase in X

Residuals or errors

• difference between an actual observed value of y and the predicted value of Y at an observed value of x

  • residuals= y_i - ŷ_i = observed value - predicted value

Sum of squared errors/ residuals (SSE)

• The sum of all the squared deviations between each data point and line of best fit

Sums of squares for residuals (SSE)

•  tells us how spread out the data are around the line of best fit

• The greater the SSE, the more spread out the data are around the line of best fit

• The line of best fit minimizes the sum of the squared residuals

  • The best fitting line has a small SSE

Small sample size

• very “spread out” around the population LOBF

When sample sizes are large

•  each sample LOBF will tend to be closer to the population LOBF. Sampling distribution of LOBF will be less spread out.

Confidence interval

95% CI for 𝛽₁ = b₁ +/- 2(seb₁)

 reject the null

we’re saying that we have strong enough evidence that there is a linear relationship between x and y on the population level

Larger R²

•  there’s a stronger linear relationship, all the points are closer to the line of best fit.