Statistics Essentials for BIOL 3000 Summary & Study Notes

🧪 Overview of Statistical Analysis

Statistics help us answer biological questions by focusing on estimation, uncertainty, and reproducibility. They are tools for describing how a system is changing, how uncertain that change is, and how confident we are that redoing the study would yield similar results.

📈 What statistical tests are doing

• Estimating how something is changing: quantify the direction and magnitude of effects.
• Estimating uncertainty: quantify how precisely we know the estimate.
• Assessing replication: gauge whether the result would be observed again under similar conditions.

🧭 Interpreting results

Results require four components that contribute to a conclusion:

1. Direction — the sign of the effect (which way the change goes).
2. Magnitude (Effect size) — how big the change is.
3. Precision (Confidence intervals) — how precisely we know the estimate.
4. Evidence of consistency of the results (test statistics) — how consistently the data support the effect.

🧬 Reporting results in biology terms

As you relate findings, be as specific as possible and describe data biologically rather than through statistics. Statistical results should supplement the narrative, not replace it. For example, if a treatment changes a dependent variable, report the magnitude and direction, with the P-value in parentheses.

• “By day 8, cowbirds reared with host young were, on average, 14% heavier than cowbirds reared alone (unpaired t = −2.23, P = 0.041, Fig. 2A).”

If the P-value exceeds 0.05, report noticeable trends rather than simply dismissing the effect as non-significant.

🧭 History of statistics

• A cup of tea started it all: Sir Ronald Aylmer Fisher.
• In 1929 Fisher argued that a 'P-value' of 0.05 is an arbitrary but convenient threshold for deciding what to ignore. He treated the P-value as a guide for what to drop, not a claim about the truth of a claim.
• He warned against overinterpreting isolated significant results and emphasized the need for reproducible designs.

American Statistical Association’s Statement on Statistical Significance and p-values

1. P-values indicate how incompatible the data are with a specified statistical model.
2. P-values do not measure the probability that the studied hypothesis is true, or that the data were produced by random chance alone.
3. Scientific conclusions should not be based solely on hitting a threshold.
4. Proper inference requires full reporting and transparency.
5. A P-value does not measure the size of an effect or its importance.
6. By itself, a P-value does not provide a good measure of evidence regarding a model or hypothesis.

🧪 What is a P-value?

A P-value is defined as the probability of obtaining test results at least as extreme as the observed result, assuming the null hypothesis is true:

$$p = P\left( \text{data at least as extreme as observed} \mid H_0 \right)$$

A very small P-value means such an extreme outcome would be very unlikely under the null model.

🔍 What do different p-values mean?

• A P-value around 0.05 is commonly treated as a threshold for “statistical significance.”
• Larger P-values (e.g., 0.1) indicate weaker evidence against the null.
• Extremely large P-values (e.g., 0.5) provide little to no reason to reject the null.

⟷ Is there a better way?

• Use a test statistic that summarizes the data under a specific model.
• Use confidence intervals to express a range of plausible values for the population parameter.

🧮 How to get Confidence Intervals

Confidence intervals quantify where the true population parameter is likely to lie.

• 68% CI and 95% CI are common choices:

• The 68% CI captures the middle 68% of samples around the mean, while the 95% CI captures the middle 95% of samples.

• Key relationships:

1. ~68% of the samples fall within 1 standard error (SE) of the sample mean.
2. 95% of the samples fall within ~1.96 SE of the sample mean.
3. The x% confidence interval is the range that contains x% of samples.
4. A standard error of 1 is equal to the standard deviation divided by the square root of the sample size.

💡 How to interpret confidence intervals

• A CI gives a range of plausible values for the true population parameter, not a claim about a single observed value.
• Narrow intervals imply more precision; wide intervals imply less precision.

How to compute the SE and CIs (basic forms)

• Standard error: $$\mathrm{SE} = \frac{s}{\sqrt{n}}$$
• 68% CI: $$\text{CI}_{68$$
• 95% CI: $$\text{CI}_{95$$

🧭 Revisit: Summary points from results interpretation

• Translate the data into biological terms; statistics are a supporting tool.
• Report magnitude and direction with uncertainty; provide P-values in parentheses when appropriate.
• Use confidence intervals to convey precision and potential biological importance.

🧬 Design thinking: Hypotheses to predictions to research projects

Hypotheses and Predictions

• Hypothesis: a plausible explanation for why something is occurring. The prediction is the expected outcome if the hypothesis is true.
• A prediction should specify the direction of the relationship between variables.

How to design a study

1. Come up with potential explanations for observed patterns. These are the hypotheses, and literature can help develop them.
2. For each hypothesis, generate at least one test that could support or refute the hypothesis.
3. For each test, create predictions of what outcomes would be expected if the hypothesis is true.
4. Determine if any test(s) will allow you to support some hypotheses while rejecting others.
5. Search the scientific literature for studies that support the different hypotheses and predictions.

Let’s repeat the steps for a second observation and topic!

Observed pattern Hypothesis 1 | Hypothesis 2 | Hypothesis 3+

Hypothesis 1 | Hypothesis 2 Hypothesis 3+

Test 1 | Test 2 | Test 3