Fundamental_skills_Statistics_for_biologists_Part3

Introduction to Statistics for Life Sciences

• Instructor: Donald Reid (glasgow.ac.uk)

Overview of Topics Covered

• Structure of a General Linear Model

  • Consideration of factor and covariate explanatory variables

  • Model output predictions

  • Reporting results including statistical metrics like degrees of freedom, Mean Sums of Squares, and F-ratio

Class Example: Analysis of Heights

Data Summary

• Various attributes recorded for each observation (Height, Age, Gender, Eye colour)

• Sample Size: 69

• Example heights data is detailed with gender and eye color listed for each individual.

Variation in Class Heights

Total Sums of Squares (TSS)

• TSS = 1380.64

What is Being Analyzed?

• Research Question: Can variation in class height be explained by biological sex?

  • Response Variable: Class height

  • Explanatory Variable: Sex (factor)

  • Hypotheses:

    • Null Hypothesis (H0): Class height not affected by sex

    • Alternative Hypothesis (Ha): Class height affected by sex

  • Model Structure: Model fit formula - Height ~ Sex

Categorical Model Analysis

Boxplot Representation

• Display of Height versus Sex visually represented.

Analysis of Variance (ANOVA) Results

Table Overview

• Response Variable: Height

• Degrees of Freedom (Df)

• Sum of Squares

  • Factor (Sex): Df = 1, Sum Sq = 146.48, Mean Sq = 146.48, F-value = 7.9519, P-value = 0.006312 (significant)

  • Residuals: Df = 67, Sum Sq = 1234.16

• Coefficients for the model analyzed:

  • Intercept: 65.0857, Std. Error: 0.6623

  • Factor (Sex) MALE: 2.9854, Std. Error: 1.0587

Fitting a Categorical Model

Interpretation

• Coefficients indicate fitted values to assess the relationship between height and sex

• Model equation: fheight = c + aM (males) + aF (females)

Modeling Approach with Covariates

Total and Explained Variation

• Introduction to covariate in model fitting

Covariate Model Structure

• Response variable vs. explanatory variables

  • Main equation: fheight = m · (Age) + c

  • Calculation of fitted values at different ages

Prediction in Linear Models

Example Calculation

• Prediction based on height values related to weight.

  • Example: If height increases by 1 foot for a dragon, its weight increases by 0.3 tons.

Reporting Results

• Importance of conveying relevant statistics: F-statistics, significance levels, p-values

• Example reporting: Discussing effects on test statistics and their significance thresholds

Degrees of Freedom (df)

Explanation

• Unique pieces of information quantifying variation

• Calculation of total and explained variation understood through df

  • For class height, TSS: 1380.64 and Total Dfs: 68

F-Ratio Calculation

Meaning and Importance

• F-ratio derived from comparing explained mean squares to residual mean squares

  • Formula Definition: F = Mean ESS / Mean RSS

Understanding P-values

Significance Contextualization

• P-value indicates the probability of observing such extreme results if the null hypothesis holds true.

Lab Report and Q&A Session

Future Tasks

• Expectation to analyze PTC data using R

Research Questions Defined

• Investigating links between genotype and other variables like Sex, Smoking preference, etc.

Conclusion

Skills Developed

• Choosing research questions, forming hypotheses, testing, reporting results, and predictive analytics for response variables.

Revision Notes: Introduction to Statistics for Life Sciences

Intended Learning Outcomes

1. Understand the statistical output of a General Linear Model (Reinforced in Data Analysis 2 lab)

2. Calculate values of your response variable for a given value of explanatory variable

3. Know how to report statistical results (Reinforced in Data Analysis 2 lab)

Structure of a General Linear Model

• Statistical Outputs:

  • Model output predictions include key statistical metrics:

    • Degrees of freedom

    • Mean Sums of Squares

    • F-ratio

Example Analysis: Class Heights

• Research Question: Can variation in class height be explained by biological sex?

• Response Variable: Class height

• Explanatory Variable: Sex (factor)

• Hypotheses:

  • Null Hypothesis (H0): Class height not affected by sex

  • Alternative Hypothesis (Ha): Class height affected by sex

  • Model Structure: Height ~ Sex

Data Summary

• Sample Size: 69

• Height Data: Variation among attributes including Age, Gender, Eye Color.

Analysis of Variance (ANOVA) Results

• Coefficients:

  • Intercept: 65.0857, Std. Error: 0.6623

  • Factor (Sex) MALE: 2.9854, Std. Error: 1.0587

Fitting a Categorical Model

• Interpretation of Coefficients:

  • Fitted values assess relationship between height and sex.

  • Model equation: fheight = c + aM (males) + aF (females)

Predicting Response Variable Values

• Modeling with Covariates:

  • Covariate Model Structure: Response variable vs. other explanatory variables.

  • Main equation for prediction: fheight = m · (Age) + c

Example Prediction Calculation

• If height increases by 1 foot, weight increases by 0.3 tons for the modeled variable.

Reporting Statistical Results

• Importance of Reporting:

  • Clearly convey relevant statistics:

    • F-statistics

    • Significance levels

    • P-values

Example Reporting Structure

• Discussing effects on test statistics and their significance thresholds.

• Understanding Degrees of Freedom (df) for quantifying variation.

Conclusion

• Competence in forming hypotheses, reporting results, and predictive analytics for response variables is developed through these concepts and tasks.

Revision Notes: Introduction to Statistics for Life Sciences

Intended Learning Outcomes

1. Understand the statistical output of a General Linear Model (Reinforced in Data Analysis 2 lab)

2. Calculate values of your response variable for a given value of explanatory variable

3. Know how to report statistical results (Reinforced in Data Analysis 2 lab)

Structure of a General Linear Model

• Statistical Outputs:

  • Model output predictions include key statistical metrics:

    • Degrees of freedom

    • Mean Sums of Squares

    • F-ratio

Example Analysis: Class Heights

• Research Question: Can variation in class height be explained by biological sex?

• Response Variable: Class height

• Explanatory Variable: Sex (factor)

• Hypotheses:

  • Null Hypothesis (H0): Class height not affected by sex

  • Alternative Hypothesis (Ha): Class height affected by sex

  • Model Structure: Height ~ Sex

Data Summary

• Sample Size: 69

• Height Data: Variation among attributes including Age, Gender, Eye Color.

Analysis of Variance (ANOVA) Results

• Coefficients:

  • Intercept: 65.0857, Std. Error: 0.6623

  • Factor (Sex) MALE: 2.9854, Std. Error: 1.0587

Fitting a Categorical Model

• Interpretation of Coefficients:

  • Fitted values assess relationship between height and sex.

  • Model equation: fheight = c + aM (males) + aF (females)

Predicting Response Variable Values

• Modeling with Covariates:

  • Covariate Model Structure: Response variable vs. other explanatory variables.

  • Main equation for prediction: fheight = m · (Age) + c

Example Prediction Calculation

• If height increases by 1 foot, weight increases by 0.3 tons for the modeled variable.

Reporting Statistical Results

• Importance of Reporting:

  • Clearly convey relevant statistics:

    • F-statistics

    • Significance levels

    • P-values

Example Reporting Structure

• Discussing effects on test statistics and their significance thresholds.

• Understanding Degrees of Freedom (df) for quantifying variation.

Conclusion

• Competence in forming hypotheses, reporting results, and predictive analytics for response variables is developed through these concepts and tasks.