• Recap week 10: Example questions
• Regression
• Regression app in R
• Lab report 7
• Today’s lab / short answer example questions

• Will be available from Friday 14th in the morning
• Due Friday 21st 4pm
• If you cannot meet the deadline, you need to apply for special consideration (SCA) on Canvas (with a good reason and proof of it)
• It must be your own work, do not plagiarise, we have dedicated software now to detect it

A Pearson correlation coefficient of 0.52 means

1. that the x variable causes the y variable to increase
2. that the x and y are positively correlated
3. that x and y are just not significantly correlated
4. that x and y are highly significantly correlated

The correlation coefficient is

1. the same as the covariance
2. a standardised version of the covariance
3. the square root of the covariance
4. none of the above

When a data set looks non-normal but I still would like to check for correlations between variables

1. I should use a Wilcoxon test
2. I should use a Pearson correlation test
3. I should use a Kendall or Spearman correlation test
4. I can use a Wilcoxon OR a Kendall test

A correlation test tests

1. whether a histogram shows a normal distribution
2. whether two groups are significantly different
3. whether two variables are correlated
4. none of the above

In a correlation analysis, if my p-value falls below 5% but in reality there is no correlation between the two variables,

1. I probably did not test whether the variables were normal
2. I might erroneously have used continuous variables
3. I commit a type II error
4. I commit a type I error

In a correlation analysis we normally have

1. two categorical variables
2. one categorical variable and one continuous variable
3. two continuous variables
4. any two variables

Assume that your waiting time for a bus is uniformly distributed between 0 and 10 minutes. What is the probability to wait at least 5 minutes or longer?

1. This cannot be calculated from the above
2. About 5 %
3. About 50 %
4. About 10 %

• Measures the dependency of one variable on another
• E.g. it tests the hypothesis ‘The body mass index of a person can be used to predict the risk for diabetes’
• A linear regression is a predictive model: e.g. it may allow us to predict diabetes risk from the body mass index of a person
• We assume causality, i.e. one variable drives the other one - unlike in correlation analysis!
• The two coefficients of a simple linear regression are the slope and the intercept
• In a linear regression, we test for the significance of the slope (only rarely for the significance of the intercept)

\(\frac{d\sum{(y - ax - b)^2}}{da} = -2\sum{x(y - ax - b)} = 0\)

We find that (without proof, see Crawley 2014, the R book):

\(a = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sum{(x - \bar{x})^2}}\)

Because the regression line by definition goes through the mean of x and y, we can work out the intercept:

\(\bar{y} = a\bar{x} + b\)

\(b = \bar{y} - a \bar{x}\)

done! confused? https://gallery.shinyapps.io/simple_regression/

• \(x\) is measured without error (for the type of regression presented here)
• The residuals are normally distributed
• Constant variance in y (no heterogeneity of variance, or ‘heteroscedasticity’)

We know how to test for normality, but we’ll have to learn how to assess heteroscedasticity

in R, we only need to type:

m1 = lm(y ~ x)
m1

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x  
    0.04133      1.02517  

and we are done!

How do we interpret these numbers?

• The intercept is what we called \(b\) and \(a\) is the slope
• If we want to predict a \(y\)-value for a given \(x\), we need to put the intercept and the slope into the formula:

\(y\)-value = \(x\) * slope + intercept

The slope (sometimes called a) and the intercept (sometimes called b) are the coefficients

Using summary(m1) (‘m1’ being your model output) additionally tests the intercept and slope for significance:

• intercept null hypothesis, using a t-statistic: ‘the intercept is not significantly different from zero’. In plain English: when x is zero, y is zero too.
• slope null hypothesis, using a t-statistic: ‘the slope is not significantly different from zero’.

• First, we see the call (the data input)
• Below, we see the statistics (min, median, max, etc.) of the residuals
• Then come the coefficients, i.e. the intercept and the slope, including their estimate, standard error, t-value and corresponding p-value.
• Note that the intercept is rarely important! The null hypothesis is that it is different from zero - rarely of interest!
• Last come the residual standard error, degrees of freedom, R-squared values (look at the adjusted R-squared value, see following slides) and an F-statistic (the last one is not important at this stage)

Photospectrometry: At concentration zero - is the absorption significantly different from zero?

R-squared value: the proportion of explained vs. total variance, corresponds to the squared correlation coefficient \(R\).

\(R = \frac{cov(x,y)}{s_xs_y} = \frac{\sum{(x-\bar{x})(y-\bar{y})}}{(n-1)s_xs_y}\)

Remember?

\(R^2\) is the proportion of variance in \(y\) explained by \(x\)

How can we visualise this?

• How do we interpret the output?
• What assumptions do we need to verify again…?

To check for normality of the residuals, we extract those from the model (m1$residuals) and use our known tools (visual, Shapiro-Wilk test)

To check the assumption of homogeneity of variance, we plot the residuals against the fitted values

• How to perform a linear regression
• How to interpret the output of a linear regression
• How to check for model assumptions
• How to report a linear regression analysis

• regression, to regress y on x
• intercept
• slope
• coefficients
• R-squared value
• heterogeneity of variance
• residuals, residual distribution
• fitted values