## Loading required package: ggplot2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

## 
## Call:
## lm(formula = weight ~ height, data = howell_clean)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.4179  -2.9406  -0.1327   2.8434  13.1366 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -50.48852    3.97234  -12.71   <2e-16 ***
## height        0.61542    0.02568   23.96   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.336 on 471 degrees of freedom
## Multiple R-squared:  0.5494, Adjusted R-squared:  0.5485 
## F-statistic: 574.3 on 1 and 471 DF,  p-value: < 2.2e-16

Simple Linear Regression

• Simple Linear Regression is a statistical model used to summarize and study statistical relationships between two continuous (quantitative) variables
• Statistical Relationships are instances in which the relationship between variables is not perfect, as in deterministic relationships

Variables Observed in a Simple Linear Regression:

• x variable, the independent variable
• y variable, the dependent variable

Examples of Linear Regression

## 
## Call:
## lm(formula = weight ~ height, data = howell_clean)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.4179  -2.9406  -0.1327   2.8434  13.1366 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -50.48852    3.97234  -12.71   <2e-16 ***
## height        0.61542    0.02568   23.96   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.336 on 471 degrees of freedom
## Multiple R-squared:  0.5494, Adjusted R-squared:  0.5485 
## F-statistic: 574.3 on 1 and 471 DF,  p-value: < 2.2e-16

3D Scatter Plot of Age, Height, and Weight

Slide with Code for Linear Regression Model

height_seq <- seq( min(howell_clean\(height), max(howell_clean\)height), length.out = 100 )

pred_df <- data.frame( height = height_seq, weight = predict(model_hw, newdata = data.frame(height = height_seq)) )

p <- plot_ly() %>% add_markers( data = howell_clean, x = ~height, y = ~weight, marker = list(size = 6, opacity = 0.6), name = “Observed Data” ) %>% add_lines( data = pred_df, x = ~height, y = ~weight, line = list(width = 2, color = “darkred”), name = “Line of Best Fit” ) %>% layout( title = “Height vs Weight”, xaxis = list(title = “Height (cm)”), yaxis = list(title = “Weight (kg)”) )

Slide with Linear Regression Model Using ggplot

## `geom_smooth()` using formula = 'y ~ x'

Slide with ggplot Code

ggplot(howell_clean, aes(x = age, y = height)) + geom_point(alpha = 0.6) + geom_smooth(method = “lm”, se = FALSE, color = “blue”) + labs( title = “Age vs Height”, x = “Age (years)”, y = “Height (cm)” ) + theme_light()

Slide with Linear Regression Model

## `geom_smooth()` using formula = 'y ~ x'

Linear Regression Model for Height and Weight

\[ \hat(weight) = \beta_0 + \beta_1 \cdot height \]

• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope of the regression line

Linear Regression Model for Height and Weight

Below is the linear regression model for height and weight based on the data taken from lgrdata:Howell:

$$ = -50.49 + 0.615 height