1.5.3: Testing hypotheses--Inferential statistics

Last updated
Save as PDF

Page ID: 75930

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

What is a hypothesis and are there different kinds?

Biological (Scientific) hypothesis: An idea that proposes a tentative explanation about a phenomenon or a narrow set of phenomena observed in the natural world. This is the backbone of all scientific inquiry! As such it is important to have a solid biological hypothesis before moving forward in the scientific method (i.e. procedures, results, discussion). After the creation of a solid biological hypothesis, it can then be simplified into a statistical hypothesis (as defined below) that will become the basis for how the data will be analyzed and interpreted.

Statistical hypotheses: After defining a strong biological hypothesis, a statistical hypothesis can be created based on what you will predict will be the measured outcome(s) (dependent variable(s)). If a study has multiple measured outcomes there can be multiple statistical hypotheses. Each statistical hypothesis will have two components (Null and Alternative).

Null hypothesis (Ho) –This hypothesis states that there is no relationship (or no pattern) between the independent and dependent variables.
Alternative hypothesis (H1) – This hypothesis states that there is a relationship (or is a pattern) between the independent and dependent variables.

Independent versus dependent variables: For both biological and statistical hypotheses there should be two basic variables defined:

Independent (explanatory) variable – It is usually what phenomena you think will affect the measure you are interested in (dependent variable).
Dependent (response) variable – A dependent variable is what you measure in the experiment and what is affected during the experiment. The dependent variable responds to (depends on) the independent variable. In a scientific experiment, you cannot have a dependent variable without an independent variable.

Example

Yellow-billed Cuckoo nests were counted during breeding season in degraded, restored, and intact riparian habitats to see overall habitat preference for nesting sites increased with habitat health.

Scientific hypothesis: Yellow-billed Cuckoo will have habitat preferences because of habitat health/status.
Statistical hypotheses: (Ho) There will be no differences in number of nests between habitats with different health/status. (H1) There will be more nests in restored and intact habitats compared to degraded.
Independent variable = Habitat health/status
Dependent variable = Number of nests counted

How do you reach conclusions?

Finally, after defining the biological hypothesis, statistical hypothesis, and collecting all your data, a researcher can begin statistical analysis. A statistical test will mathematically “test” your data against the statistical hypothesis. The type of statistical test that is used depends on the type and quantity of variables in the study, as well as the question the researcher wants to ask. After computing the statistical test, the outcome will indicate which statistical hypothesis is more likely. This, in turn indicates to scientists what level of inference can be gained from the data compared to the biological hypothesis (the focus point of the study). Then a conclusion can be made based on the sample about the entire population. It is important to note that the process does not stop here. Scientists will want to continue to test this conclusion until a clear pattern emerges (or not) or to investigate similar but different questions.

Types of Basic Statistical Tests

Inferential statistics generally provide a test statistic, the degrees of freedom (related to the number of individuals in each sample) and a p-value. Significance (acceptance of the alternative hypothesis) is generally based on the p-value. Depending on the field, scientists will often use a cut-off of 0.01 or 0.05 to determine significance. If the test returns a p-value that is less than this value, the relationship is deemed significant.

Chi-Square – Are two categorical variables related?
- (e.g. do different habitats different in the numbers of species of each type?)
T-Test – Does the mean (continuous data) of one group (a categorical variable) differ from the mean of another group?
- (e.g. are oak trees taller than hickory trees, on average?)
ANOVA – Does the mean of several groups differ? A post-hoc test is used to run pairwise comparisons if so
- (e.g. does height differ across tree species, on average?)
Linear regression - Are two continuous variables linearly related?
- (e.g. do taller trees have a larger circumference?)

The “Magic” level of Significance

If p ≤ 0.05 – accept alternative hypothesis

There is less than 5% chance that the samples are from the same population
There is a significant difference between the samples

If p > 0.05 – accept null hypothesis

There is no significant difference between the samples

Attribution

Rachel Schleiger (CC-BY-NC)