Introduction and Series Context

• Part 2 of General Linear Models series from StatQuest
• Presented by genetics department at University of North Carolina at Chapel Hill
• Previous episode covered linear regression, this episode extends same techniques to t-tests and ANOVA
• Uses design matrix concept for unified approach

Linear Regression Review

• Previous example: mouse weight vs mouse size analysis
• Two key questions answered:
  • How useful is mouse weight for predicting size (r-squared)
  • Is relationship due to chance (p-value)
• Goal: apply same concepts to t-tests

T-Test Application Setup

• Example: gene expression comparison between control mice and mutant mice
• Mutant mice = normal mice with specific gene knocked out/non-functioning
• T-test goal: compare means and determine if significantly different
• Same method for calculating p-values enables handling more complex situations

Step-by-Step T-Test Process Using Linear Regression Framework

Step 1: Find Overall Mean

• Ignore x-axis initially, focus on y-axis
• Calculate overall mean for both linear regression and t-test cases
• X-axis labels removed to emphasize y-axis focus

Step 2: Calculate Sum of Squared Residuals Around Mean (SS_mean)

• Calculate residuals (distance from data points to overall mean line)
• Square and sum these residuals
• Same process for both regression and t-test

Step 3: Fit Lines to Data

• Linear regression: single least squares fit line
• T-test approach:
  • Fit separate lines to control and mutant data
  • Control data: mean = 2.2, equation y = 2.2 (horizontal line)
  • Mutant data: mean = 3.6, equation y = 3.6 (horizontal line)
  • Two separate horizontal lines (means are least squares fits)

Step 4: Combine Lines Using Design Matrix

• Challenge: combine two separate lines into single equation for computer automation
• Unified equation format: y = (coefficient₁ × control_mean) + (coefficient₂ × mutant_mean) + residual
• Control points: coefficient₁ = 1, coefficient₂ = 0
• Mutant points: coefficient₁ = 0, coefficient₂ = 1
• Coefficients act as on/off switches for respective means

Design Matrix Structure

• Matrix of 1s and 0s isolated from equations
• Column 1: controls control mean (on/off)
• Column 2: controls mutant mean (on/off)
• Combined with abstract equation: y = control_mean + mutant_mean
• Enables single equation representation of fit

Step 5: Calculate Sum of Squares Around Fitted Lines

• Linear regression: sum of squared residuals around fitted line
• T-test: sum of squared residuals around fitted means (horizontal lines)

Step 6: Calculate F-statistic and P-value

• Use same F equation for both regression and t-test
• Parameters:
  • p_mean = 1 (single parameter for overall mean in both cases)
  • p_fit for regression = 2 (intercept + slope)
  • p_fit for t-test = 2 (control mean + mutant mean)
• F-statistic leads to p-value calculation

ANOVA Extension

• Tests if all categories have same mean
• Example: 5 categories total
  • Control mice
  • Mutant mice
  • Control mice on funky diet
  • Mutant mice on funky diet
  • Heterozygous mice

ANOVA Process

• Step 1: Calculate sum of squares around overall mean (same as before)
  • Overall mean equation: y = mean_expression
  • p_mean = 1 (single parameter)
• Step 2: Calculate sum of squares around fitted lines
  • Equation has 5 parameters (one mean per category)
  • p_fit = 5
• Design matrix: one column per category (5 columns total)
• Calculate F-statistic and p-value using same framework

Design Matrix Variations

• StatQuest used specific design matrix format
• More common alternative design matrix exists for same t-test
• Both matrices achieve same results
• Standard matrices will be covered in next episode
• Future episodes will cover more elaborate designs