Correlation and Regression Lecture 5 Objectives Outline: - - PowerPoint PPT Presentation
Correlation and Regression Lecture 5 Objectives Outline: Situations when our explanatory variable is more continuous than categorical By actively following the lecture and practical and carrying out the independent study the successful
Objectives
Outline: Situations when our explanatory variable is more continuous than categorical By actively following the lecture and practical and carrying out the independent study the successful student will be able to:
both in R
scientifically
Choosing tests…
Test of differences
Explanatory variables: discrete Response variable: continuous Test: t-tests, anova
Test of relationship
Explanatory variables: continuous Response variable: continuous Test: regression/correlation
Correlation and Regression
Related but DIFFERENT techniques Correlation – association Regression – predictive relationship Linear (always do a scatterplot first!)
Correlation
Regression
and response)
Correlation vs regression
Correlation coefficients
Moment
■
Denoted by ‘r’
■
Ranges from -1 to +1
Types of correlation
Highest scores on one axis associated with highest scores on other
Lowest scores on one axis associated with highest scores on other
Types of correlation
Types of correlation
No correlation!
No LINEAR correlation!
x = variable 1 y = variable 2 n = sample size
Correlation: example
20 leaf samples: 10 compounds measured Investigate correlation between c1 and c2 ALWAYS DO A SCATTER PLOT
Correlation: example
Investigate normality Not great but we will carry on for demonstration purposes
Correlation: Example
Running and interpreting the test
cor.test(comp$c2, comp$c1, method = " pearson") data: comp$c2 and comp$c1 t = 2.3132, df = 18, p-value = 0.03274 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.04589555 0.76018365 sample estimates: cor 0.4786942
The default method = c("pearson", "kendall", "spearman") r could be between 0.046 and 0.76 A wide margin, but positive Correlation coefficient, r t-test of whether r is different from zero
Correlation: Example
Reporting the result data: comp$c2 and comp$c1 t = 2.3132, df = 18, p-value = 0.03274 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.04589555 0.76018365 sample estimates: cor 0.4786942
There was a significant positive correlation between compounds 1 and 2 (r = 0.48; t = 2.31; d.f. = 18; p = 0.033). Significance Direction Statistics
Correlation: Example
Understanding the significance test A t-test and works in the same way as all t-tests. Problem: Sensitive to sample size: Big n -> small s.e -> big t -> small p
Correlation: Example Consider both the p value and the r value
Correlation - nonparametric alternative
Not sure about normality? Spearman’s Rank Correlation
Running and interpreting the test
cor.test(comp$c2, comp$c1, method = "spearman") Spearman's rank correlation rho data: comp$c2 and comp$c1 S = 664, p-value = 0.02605 alternative hypothesis: true rho is not equal to 0 sample estimates: rho 0.5007519
Regression
Prediction One variable (the explanatory) causes the other (the response) Develops a best fitting straight line: ! = b1" + b0 b1 = gradient b0 = y intercept
Regression
! = b1" + b0 !i - the observed y for "i ŷi - the predicted y for "i Residual: !i - ŷi Best fitting: minimising sum of squared residuals
Regression
Null hypothesis can be expressed as:
But all mean the same
Regression: example
Concentration of juvenile hormone (JH) and mandible length in stag beetles ALWAYS DO A SCATTER PLOT
Regression: Example
Running the test
mod <- lm(data = stag, mand ~ jh) summary(mod) response ~ explanatory
Interpreting the test
Regression: Example
Summary statistics about residuals
Interpreting the test
b0 (y intercept) b1 (gradient)
Regression: Example
t-test of b0 = 0 Often not impt t-test of b1 = 0 Always of interest
Test of ‘model’ Same as t-test
for single regression
Interpreting the test
Regression: Example
Proportion of y explained by x
Regression: Example
Reporting the result
The concentration of juvenile hormone explained a significant amount of the variation (0.54) in stag beetle mandible length (F = 16.6; d.f. = 1,4; p = 0.00113). The regression line is mandible length = (0.032*Jhconc) +0.419. Significance Direction Statistics
Regression: Example
Illustrating result Include the data and the line (the model)
ggplot(data = reg,aes(x = QTL, y = pheno)) + geom_point(size=2) + xlim(0,15) + ylim(0,40) + xlab("Number of QTL") + ylab("Percentage of phenotype") + geom_smooth(method = "lm", se = FALSE) + theme_bw()
Regression: Example
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plot(mod) Not that useful here
Spread should be similar in each group: equal variance Should be approx 1:1 for normality
Checking assumptions after running the regression Use the residuals
Regression: Example
Predicting from the model
Summary
Both
Correlation
Regression
and possibly r2
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Objectives
Outline: Situation when our explanatory variable is more continuous than categorical By actively following the lecture and practical and carrying out the independent study the successful student will be able to:
both in R
scientifically