Lecture 7 - Hypothesis Testing & Linear Modelling

Scientific Hypotheses

Expected or predicted relationships between two or more variables that motivate a piece of research. Stems from a research question and are often developed based on prior knowledge or theory.

Statistical Hypotheses

Restating a research hypothesis so that it can be addressed by statistical techniques through hypothesis testing. Includes a:

• Null hypothesis: no news here, there’s no different, there’s no relationship, there’s nothing going on, there’s nothing different from before. This is the hypothesis actually being tested

• Alternative hypothesis: new news, something is different, there is something going on, there is some sort of relationship. Contains possibilities more biologically interested and contains values that are predicted, which we’ll use to test against the null

Four Steps of Hypothesis Testing

1. State the hypotheses

2. Compute the relevant test statisation (summary of dataset used to evaluate how compatible it is to the expected/null model)

3. Determine the p-value

4. Draw the appropriate conclusions

Before stating the hypotheses, we also have to decide what statistical test to use based on the metric we wish to investigate, the nature of our variables (categorical, numerical, etc), and assumptions of the test

One Sample t-Test

Requires one variable and one sample

• The null would state that the metric would equal some sort of known value (average of a plant species height is 5cm, etc)

• The alternative would state that the metric isn’t equal to that known value (average of plant species height is not 5cm, etc)

Normally, we’d have to go through the steps of manually calculating everything we’d need to find the p-value (standard error of mean, then the t-statistic, then using the t-distribution to find our p-value.

However, we can use R code to do it for us with

• t.test(name_of_dataset$response_variable, mu = null_value)

This gives us everything we need to draw an appropriate conclusion

Two Sample t-Test

The steps are very similar to the one sample t-test

First, we’d state our hypotheses.

• The null would state that there is no difference between the two groups or the slope is zero

• The alternative would state there is some sort of difference between the two groups or the slope isn’t zero

Then, we’d use R code to calculate the t-statistic and find the p-value for us with:

• lm(response_variable ~ explanatory_variable, data = dataset)
  summary(lm_object)$coef

or

• t.test(response_variable ~ explanatory_variable, data = dataset)
  summary(lm_object)$coef

With all this information, we’d derive an appropriate conclusion

Drawing Appropriate Conclusions

An appropriate conclusion would include:

• The sample size

• The test statistic

• The explicit p-value

• What the p-value means in terms of evidence for/against the null

• Confidence intervals to show the size of the effect

An example would be:

When comparing bill length in Gentoo (n = 57) and Adèlie (n = 73), we found very strong evidence against the null hypothesis of there being no difference in average bill length between these two species (two sample t-test, t = 3.25, p = 0.002)