Inferential Statistics Overview

Inferential Statistics Overview

• Inferential statistics uses samples to draw conclusions about larger populations.
• It's a tool to understand the underlying nature of phenomena.
• Inferential statistics allows researchers to test hypotheses and determine relationships between variables.
• It enables predictions about populations based on sample data.

Learning Outcomes

• Students will be able to draw reliable conclusions from samples of large populations.
• Students will be able to compare different populations.
• Students will be able to explain when to use various inferential statistical methods.
• Students will be able to perform calculations involved in t-tests and Chi-Squared tests.

Hypothesis Testing

• A formalized method for understanding the world.
• It compares a hypothesis of interest with alternative hypotheses.
• The hypothesis of interest states a specific relationship between variables.
• The null hypothesis states there is no relationship between variables.

Relationship Between Variables

• Correlation coefficients describe the linear relationship between two variables.
  • Correlation coefficients range from -1 to 1.
  • A value of -1 indicates a strong negative correlation.
  • A value of 1 indicates a strong positive correlation.
  • A value of 0 indicates no linear correlation.
• Visualizations like scatter plots illustrate the correlation between variables.

Hypothesis Testing (cont.)

• The p-value represents the probability of observing the data if the null hypothesis is true.
• The alpha value is a pre-determined threshold for rejecting the null hypothesis.
• A p-value less than alpha indicates rejection of the null hypothesis.
  • Common alpha values are 0.05 and 0.01.

Hypothesis Testing (cont.)

• Type 1 error (false positive): rejecting a true null hypothesis.
• Type 2 error (false negative): failing to reject a false null hypothesis.
• Alpha value influences the probability of a Type 1 error.
• Statistical power relies on sample size and effect size to avoid Type 2 errors.

Tests of Significance

• Nonparametric: Chi-Square, Fisher's exact test - Analyse relationship between categorical variables
• Parametric: T-test (paired, unpaired), ANOVA, Pearson's Correlation, Linear Regression - Used when data are normally distributed - Compare means/variances in groups

Parametric Tests

• Used when data are normally distributed.
• Compare means and variances.
• The Kolmogorov-Smirnov test assesses normality of data for parametric use.

Commonly Used Parametric T-Tests

• Used to compare means in two groups.
• Dependent variable: interval/ratio.
• Independent variable: nominal/ordinal (with two groups).
• Associated plots: Bar, Box, Violin.

Commonly Used Parametric ANOVA

• Used to compare means in more than two groups.
• Dependent variable: interval/ratio.
• Independent variable: nominal/ordinal (with more than two groups).
• Associated plots: Bar, Box, Violin.

Commonly Used Parametric Correlation

• Measures the dependence between two variables.
• Variables: interval/ratio.
• Associated plot: Scatter Plot

Commonly Used Parametric Regression

• Measures dependence between two or more variables.
• Dependent variable: interval/ratio.
• Independent variables: of any type.
• Associated plots: Scatter Plot, Box Plot.

Non-Parametric Tests

• Used when data don't satisfy parametric test requirements.
• Typically compare medians.
• Rank data instead of raw values.

Specific Non-parametric and Parametric tests and conditions

• Independent measures, 2 groups: Mann-Whitney test(non), independent measures t-test (param)
• Independent measures, >2 groups: Kruskal-Wallis test(non), one way independent measure ANOVA(param)
• Repeated measures, 2 conditions: Wilcoxon test (non), matched pair t-test(param)

T-Tests

• Key tool for comparing means in biological systems.
• Experimental design controls variables.
• Tests the probability that samples come from one population with a single mean.
• T-values indicate significance levels.

T-test Tables

• T-values in tables show significance level differences between two means based on sample size and t-score.
• Reject the null hypothesis if the calculated t-value is greater than the critical value in the table.

Example: T-test application

• Comparison of egg masses from different species.
• Calculated t-score greater than critical value (significance level).

One-tailed vs Two-tailed Tests

• One-tailed: Hypothesis predicts direction of effect.
• Two-tailed: Hypothesis doesn't specify direction of effect.

One-Sample t-test

• Compares a single sample mean with a known population mean.
• Calculates how many standard errors the sample mean deviates from the population mean.

Paired t-test

• Compares means of the same variable measured under different conditions or at different times.
• Divides the mean difference by the standard error of the difference.

Unpaired t-test

• Compares means of the same variable in two different samples.
• Divides the difference between sample means by the standard error of the difference.

Analysis of Variance (ANOVA)

• Compares multiple samples, avoiding multiple comparisons errors.
• Compares mean variation within samples to mean variation between samples.

One-way ANOVA

• Compares means of more than two samples.

Repeated Measures ANOVA

• For repeated measurements on the same sampling unit.

F-value in ANOVA

• ANOVA test statistic that compares between-sample and within-sample variations.
• Ratio of mean square between groups to mean square within groups.

Chi-squared Test

• Compares frequencies between expected and observed values of nominal data.

Calculating Chi Squared

• Formula for calculating Chi-squared: x² = ∑ ( (O−E)² / E ) where:
  • O = observed frequencies
  • E = expected frequencies
  • ∑ = sum of

Chi-squared Significance Level

• The significance level in Chi-squared tests depends on degrees of freedom.
• Degrees of freedom = (number of rows - 1) * (number of columns - 1)