Dyr og Data

Learning Objectives

By the end of this activity, you will be able to:

• Fit and interpret a simple linear regression model

• Understand slope and intercept in a biological context.

• Use the regression model for prediction.

• Assess model fit using R² and diagnostic plots

Context

In animal production, body weight is an important indicator of health, growth, and productivity. However, weighing large animals like cattle or sheep can be time-consuming or impractical. Body measurements (like heart girth or body length) are often used to estimate weight using regression models

Rocha-Silva et al. Trop Anim Health Prod 56, 42 (2024)

Cattle body measurments

“girth” -> “omkreds”

Cattle body measurments

Tasks

1. Plot weight_kg vs. heart_girth_cm

2. Describe the trend (positive, linear, strength, etc.).

3. Discuss: Why might heart girth be a good predictor of weight?

Equation for a straight line

At school you probably learned this (or something like it)

\[ y = m x + b \]

Schematic

Model fitting

We fit linear models using lm()

m <- lm(
  weight_kg ~ heart_girth_cm,
  data = cattle
)

look at the model estimates

m

Call:
lm(formula = weight_kg ~ heart_girth_cm, data = cattle)

Coefficients:
   (Intercept)  heart_girth_cm  
      -232.670           3.974  

Prediction

Use the model to predict the weight of an animal with the following heart_girth_cm:

1. 175cm
2. 135cm
3. 195cm

Compare predicted and observed values for these animals

predict()

In R we can use predict() to do the calculations

predict_data <- data.frame(heart_girth_cm = c(175, 135, 195))
predict(m, newdata = predict_data)

       1        2        3 
462.7006 303.7587 542.1716 

Model summary

summary(m)

Call:
lm(formula = weight_kg ~ heart_girth_cm, data = cattle)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.9652  -5.2152   0.9026   5.7848  10.4265 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -232.6704    19.6128  -11.86 3.25e-07 ***
heart_girth_cm    3.9735     0.1169   34.00 1.14e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.403 on 10 degrees of freedom
Multiple R-squared:  0.9914,    Adjusted R-squared:  0.9906 
F-statistic:  1156 on 1 and 10 DF,  p-value: 1.145e-11

Model diagnostics

gglm(m)

Linear regression

We estimate the two unknown parameters in the model using a procedure known as least squares, where we minimise the Residual Sum of Squares \(\mathrm{RSS} = \sum_{i=1}^n (y_i - \hat{y}_i)^2\)

Linear regression

Estimates of parameters (\(\beta_j\)) are for the population based on the fit to our sample of data

Linear regression

Data were 20 observations generated from the following model

\[\mu_i = 0.7 + 0.8x_i \;\;\;\; y_i \sim N(\mu_i, \sigma = 1)\]

Fitted model estimates are: \(\hat{\beta}_0\) = 0.32 and \(\hat{\beta}_1\) = 0.999

The parameters are means & the uncertainty in the estimated values is captured by their standard errors

Confidence intervals for the estimates:

• \(\beta_0 \pm t_{0.975} \mathrm{SE}_{\beta_0}\) = -0.401, 1.041
• \(\beta_1 \pm t_{0.975} \mathrm{SE}_{\beta_1}\) = 0.741, 1.256