Class 20: Many variables in linear models

Methodology of Scientific Research

Andrés Aravena, PhD

April 13, 2023

Dataset

For this part we will use the following data

https://dry-lab.org/static/students.txt

These are self-reported values from former students

Load in your spreadsheet and plot

  • weight_kg versus height_cm
  • weight_kg versus hand_span

First two models

We want two models for weight_kg

  • One based in height_cm \[\text{weight_kg} = β_0 + β_1⋅\text{height_cm}\]
  • One based in hand_span \[\text{weight_kg} = β_0 + β_1⋅\text{hand_span}\]

Obviously, \(β_0\) and \(β_1\) on each model are independent. This time we can use SLOPE() and INTERCEPT(). Please do it now

Which one is better?

We have tools to evaluate the quality of the model

Please use them to evaluate both models

Also, evaluate the sum of squared errors for each model

Another Spreadsheet function

Besides SLOPE() and INTERCEPT(), there is another function

We can get both at the same time using LINEST()

This is an advanced function that returns several values

It has some optional parameters that we will study later

Please use it and compare with the previous results

Combining both models

We have two ways to predict weight_kg

We want to combine them

\[\text{weight_kg} = β_0 + β_1⋅\text{hand_span} + β_2⋅\text{height_cm}\]

Now we have to find three parameters

What are the conditions for these parameters?
How can we find them?

Finding all model’s parameters

The formula for \((β_0, β_1, β_2)\) is complex. We will not teach it

Instead, we will use the LINEST() function

This is the main advantage of LINEST()

LINEST() can model using several independent variables

Exercise

Evaluate Mean Squared Error and \(R^2\) for the large model

Which one is the better now?

Modeling free fall

Example videos

YouTube has an option to show the video at 0.25 speed. Try it

Summary of the video

We analyzed the video

We got time and position from the video

You can get the data at
https://dry-lab.org/static/2020/cmb1/free-fall.csv

There is another version using comma as decimal separator
https://dry-lab.org/static/2020/cmb1/free-fall-eu.csv

Read them in your spreadsheet

Looking for the straight line

Plot the data directly

Then in semi–log scale

Then in log–log scale

Which one would you choose?

This is not any exponential

There are many possible functions that connect \(x\) and \(y\)

We will try a polynomial formula

A polynomial is a formula like \[y=β_0 + β_1\cdot x + β_2\cdot x^2 +\cdots+ β_n\cdot x^n\]

The highest exponent is the degree of the polynomial

We will try a 2-degree formula \[y = β_0 + β_1\cdot x + β_2\cdot x^2\]

We will use a polynomial model

\[\text{distance}=β_0 + β_1\cdot\text{time}+β_2\cdot\text{time}^2\] This is a parabola, which can be modeled with a second degree polynomial

When we use a polynomial model, we need to add new columns

In this case we need to add time_sq containing time2

This is used only for the model

Time starts at zero

  • In the video the timer starts before dropping the ball
  • The first data is when the ball crosses 10
  • We choose this as our time reference
  • The ball is already moving. Speed is not zero
  • Initial speed is not zero

\(β_0\) should be 0

Since we choose that at time=0 we have distance=0, we know that \(β_0\) is 0

In other words, our model here is \[y =β_1\cdot x + β_2\cdot x^2\]

The old \(β_0\) has a fixed value of 0

How do we do that in Excel?

Meaning of the coefficients

If you remember your physics lessons, you can recognize that in the formula \[y =β_1\cdot x + β_2\cdot x^2\] the value \(B\) is the initial speed, and \(β_2=-g/2\)

Thus, we can measure the gravitational acceleration in the second coefficient