Exercise 1: Gestational age and birthweight Draw a line of best fit through the data (with roughly half the points above and half below). Describe the relationship Is the relationship: strong/ weak? positive/ negative? linear?

Exercise 2: Interpretation Interpret the following correlation coefficients using Cohen’s classification and explain what they mean. Which correlations seem meaningful? Relationship Correlation Average IQ and chocolate consumption 0.27 Road fatalities and Nobel winners 0.55 Gross Domestic Product and Nobel winners 0.70 Mean temperature and Nobel winners -0.60

Exercise 3 Investigate whether mother’s pre-pregnancy weight and birth weight are associated using a scatterplot, correlation and simple regression

Exercise 3: correlation Pearson’s correlation coefficient = Describe the relationship using the scatterplot and correlation coefficient

Exercise 3: regression Adjusted R2 = Does the model result in reliable predictions? ANOVA p-value = Is the model an improvement on the null model (where every baby is predicted to be the mean weight)?

Exercise 3: Regression Pre-pregnancy weight coefficient and p-value: Regression equation: Interpretation:

Exercise 4: correlations Which variables are most strongly related to each other?

Exercise 5 Run a multiple regression model for Birth weight with gestational age, mothers weight and smoking status as independent variables Check the assumptions and interpret the output Does the model give more reliable predictions than the model with just gestational age? Add mother’s height to the model. Does anything change? Note you will need to create a variable for smoking status based on the number of cigarettes that the mother smokes (assuming that 0 cigarettes indicates someone who does not smoke)

Exercise 5: model 1 summary Variable Coefficient (β) P-value Significant? Constant Gestation Smoker Pre-pregnancy weight Adjusted R2 = Interpretation:

Exercise 5: model 2 summary Variable Coefficient (β) P-value Significant? Constant Gestation Smoker Pre-pregnancy weight Height Adjusted R2 = Interpretation:

Exercise 5: Compare p-values Model Gestation Smoking Weight Height P < 0.001 + Smoker 0.028 + Weight + Height

Exercise 5: Compare R2 Model R2 Adjusted R2 Gestation 0.499 0.486 + Smoker 0.558 0.535 + Weight + Height

Exercise 1: Gestational age and birthweight There is a strong positive relationship which is linear

Exercise 2: Interpretation Relationship Correlation Interpretation Average IQ and chocolate consumption 0.27 Weak positive relationship. More chocolate per capita = higher average IQ Road fatalities and Nobel winners 0.55 Strong positive. More accidents = more prizes! Gross Domestic Product and Nobel winners 0.7 Strong positive. Wealthy countries = more prizes Mean temperature and Nobel winners -0.6 Strong negative. Colder countries = more prizes.

Exercise 3: scatterplot Assumption 1: Is there a linear relationship? Yes!

Exercise 3: correlation Pearson’s correlation = 0.40 Describe the relationship using the scatterplot and correlation coefficient: There is a moderate positive relationship between mothers’ pre-pregnancy weight and birth weight (r = 0.40). Generally, birth weight increases as mothers weight increases

Exercise 3: regression Adjusted R2 = 0.14 Does the model result in reliable predictions? Not really. The adjusted R2 value is 0.14. ANOVA p-value = 0.009 Is the model an improvement on the null model (where every baby is predicted to be the mean weight)? Yes as p < 0.05

Exercise 3: regression Pre-pregnancy weight coefficient & p-value: 0.034 (p = 0.009) Regression equation: y = 1.379 + 0.034 Interpretation: There is a significant relationship between a mothers’ pre-pregnancy weight and the weight of her baby (p = 0.009). Pre-pregnancy weight has a positive affect on a baby’s weight with an increase of 0.034 kg for each extra kg a mother weighs.

Exercise 3: normality of the residuals? Yes – histogram roughly peaks in the middle

Exercise 3: homoscedasticity? Yes – no patterns in residuals

Exercise 4: correlations Which variables are most strongly related? Gestation and birth weight (0.708) Mothers height and weight (0.681) Mothers height and weight are strongly related. They don’t exceed 0.8 but try the model with and without height in case it’s a problem

Exercise 5: Model 1 residual assumptions Assumptions are met

Exercise 5: Model 1 Adjusted R2 Does the model result in reliable predictions? Yes – the adjusted R2 is reasonably high

Exercise 5: Model 1 ANOVA ANOVA p-value < 0.001 Is the model an improvement on the null model (where every baby is predicted to be the mean weight)? Yes as p < 0.05

Exercise 5: Model 1 regression coefficients Gestational age (p < 0.001), mother’s pre-pregnancy weight and smoking status (p=0.016) are significant predictors of birth weight after controlling for the weight and height of the mother. Weight increases by 0.14 kgs for each week of gestation, by 0.02 kgs for each extra kg a mother weighs and decreases by 0.31 kgs for smokers.

Exercise 5: Model 2 (including height) There’s a moderate positive linear relationship between maternal height and birth weight.

Exercise 5: Model 2 residual assumptions Assumptions are met

Exercise 5: Model 2 Adjusted R2 Does the model result in reliable predictions? Yes – the adjusted R2 is reasonably high

Exercise 5: Model 2 ANOVA ANOVA p-value < 0.001 Is the model an improvement on the null model (where every baby is predicted to be the mean weight)? Yes as p < 0.05

Exercise 5: Model 2 regression coefficients Interpretation: Gestational age (p < 0.001) and smoking status (p=0.016) are significant predictors of birth weight after controlling for the weight and height of the mother. Weight increases by 0.14 kgs for each week of gestation and decreases by 0.31 kgs for smokers.

Exercise 5: compare p-values Model Gestation Smoking Weight Height P < 0.001 + Smoker 0.024 + Weight 0.016 0.025 + Height 0.263 0.366 Smoking gets more significant as variables are added. Mothers’ weight becomes non-significant once height has been added. They are strongly related and are sharing some of the variation in birth weight when both in the model.

Exercise 5: compare Adjusted R2 Model R2 Adjusted R2 Gestation 0.502 0.489 + Smoker 0.563 0.541 + Weight 0.618 0.588 + Height 0.627 0.586 Adding smoker and weight improves the fit a little bit Adding height has not improved the fit of the model at all as the adjusted R2 decreases