Week 05: Simple Regression with JAMOVI

Date: September 24, 2026

Today…

The Linear Model — the same machine, driven by point-and-click

Why a line? (and why regression first)
Data = Model + Error
Where the intercept and slope come from
Reading a slope out loud
How well does the line fit? ($R^2$, effect size)
The punchline from Lab 5

Note

This is the JAMOVI version of this week’s lecture. The concepts, the data, and every number are identical to the R version — only the buttons change. Use whichever you like; the exam-room skill is reading the output, not producing it.

First, a confession about Lab 5

In Lab 5 you ran three things:

A t-test comparing two groups
A correlation between two continuous variables
An lm() — the “sneak peek” 👀

And the numbers came back the same.

That was not a bug, and it was not a coincidence.

Important

Regression is not a new test that sits next to t-tests and correlations. It is the general machine that both of them are special cases of. In JAMOVI, that machine is Analyses → Regression → Linear Regression.

Why we teach regression first

Regression is a general data analytic system

Lots of things fall under the umbrella of regression
It handles many forms of relations and types of variables
The output gives you effect sizes and statistical significance
We can add multiple predictors and account for their intercorrelations

Learn one machine well, then add parts to it. That is the whole plan for the next eight weeks.

Uses for regression

Adjustment: Take into account (control) known effects in a relationship
Prediction: Develop a model based on what has happened previously to predict what will happen in the future
Explanation: Examine the influence of one or more variables on some outcome

Part 1: Building the Line

The core idea

With regression, we are building a model that we think best represents the data, and the broader world

\[ Data = Model + error \]

Every model we build for the rest of the semester is a variation on this sentence.

A line you already know

At the simplest form we draw a line to characterize the linear relationship between two variables, so that for any value of x we have an estimate of y

\[ Y = mX + b \]

Y = Outcome Variable (DV)
m = Slope Term

X = Predictor (IV)
b = Intercept

Let’s “science up” the equation

\[ Y_i = b_0 + b_1X_i + e_i \]

This captures how we calculate each observation ( $Y_i$ )

\[ \hat{Y_i} = b_0 + b_1X_i \]

This gives the “best guess” or expected value of $Y$ given $X$

Expected vs. Actual

\[Y_i = b_{0} + b_{1}X_i + e_i \qquad \hat{Y_i} = b_{0} + b_{1}X_i\]

$\hat{Y}$ signifies the fitted score — no error. We interpret it as “on average.”

The difference between the fitted and the observed score is the residual:

\[Y_i - \hat{Y_i} = e_i\]

Note

JAMOVI will hand you both of these if you ask: in Linear Regression → Save, tick Predicted values and Residuals, and two new columns appear in your spreadsheet.

OLS

How do we find the regression estimates?
Ordinary Least Squares estimation
Minimizes deviations
- \[ min\sum(Y_{i} - \hat{Y} ) ^{2} \]
Other estimation procedures are possible (and necessary in some cases)

OLS

The line that yields the smallest sum of squared deviations

\[\Sigma(Y_i - \hat{Y_i})^2\] \[= \Sigma(Y_i - (b_0+b_{1}X_i))^2\] \[= \Sigma(e_i)^2\]

You could try many different coefficients until you find the pair with the smallest sum of squared deviations. Luckily, there are simple calculations that yield the OLS solution every time — and JAMOVI does them the moment you drop a variable in a box.

🔍 Watch OLS happen

Drag the points around and watch the line chase them:

https://setosa.io/ev/ordinary-least-squares-regression/

https://observablehq.com/@yizhe-ang/interactive-visualization-of-linear-regression

Where does the slope come from? 🤔

Remember covariance from last week:

\[cov_{xy} = \frac{\sum(x-\bar{x})(y-\bar{y})}{N-1}\]

and the correlation, which standardizes it:

\[r_{xy} = \frac{cov_{xy}}{s_x s_y}\]

The regression coefficient is built from exactly the same pieces.

Regression coefficient, $b_{1}$

\[b_{1} = \frac{cov_{XY}}{s_{x}^{2}} = r_{xy} \frac{s_{y}}{s_{x}}\]

What units is the regression coefficient in?

The slope equals the estimated change in Y for a 1-unit change in X

Tip

A correlation is unitless. A slope is in the units of your study. That is why a slope is often the more useful number to report to a human being.

Slope is the correlation (rescaled)

\[\Large b_{1} = r_{xy} \frac{s_{y}}{s_{x}}\]

If the standard deviation of both X and Y is equal to 1:

\[\Large b_1 = r_{xy} \frac{s_{y}}{s_{x}} = r_{xy} \frac{1}{1} = r_{xy} = \beta_{yx}\]

Standardize both variables and the slope becomes the correlation. In JAMOVI that is the Standardized estimate checkbox under Model Coefficients — tick it and watch it equal r.

Estimating the intercept, $b_0$

The intercept adjusts for differences in the means of X and Y

\[\hat{Y_i} = \bar{Y} + r_{xy} \frac{s_{y}}{s_{x}}(X_i-\bar{X})\]

It is where the regression line crosses the y-axis, at $X = 0$
It ensures the line runs through the point $(\bar{X}, \bar{Y})$

\[\Large b_0 = \bar{Y} - b_1\bar{X}\]

Part 2: Reading a Line Out Loud

Our data today 🃏

Cards Against Humanity: Pulse of the Nation — a national poll that asked serious demographic questions and some very strange ones.

Download the data (.csv)

☰ Menu > Open > Browse > CAH.csv

JAMOVI will guess the measure type for each column. Check the icons at the top of each column:

age, transformers, books should be continuous (ruler icon)
gender should be nominal (three circles)

Click the column header to fix any it guessed wrong.

Step 1: ALWAYS visualize first

We need the scatr module for scatterplots.

➕ Modules > jamovi library > install scatr

Then:

Analyses > Exploration > scatr > Scatterplot

X-Axis: age · Y-Axis: transformers
Regression Line: choose Linear

Talk about it (2 min)

The line slopes down. Before we fit anything: how strong does that relationship look to you? If you had to guess r, what would you say? Write your guess down.

Step 2: Descriptives first

Analyses > Exploration > Descriptives

Drag age and transformers into Variables.

Variable	N	Mean	SD	Min	Max
`age`	1000	49.41	16.63	18	99
`transformers`	938	1.33	1.56	0	5

Warning

Notice the two different Ns. 62 people did not answer the Transformers question. JAMOVI, like R, will quietly use only the complete cases when it fits the model. Worth knowing before you report an N.

Step 3: The correlation

Analyses > Regression > Correlation Matrix

Drag in age and transformers. Tick Pearson, p-value, and (optionally) Correlation matrix under Plot.

	age	transformers
transformers	r = −0.266	—
	p < .001

How close was your guess?

Step 4: The same relationship as a line

Analyses > Regression > Linear Regression

Dependent Variable: transformers
Covariates: age (continuous predictors go in Covariates)

Open Model Coefficients and tick Standardized estimate.

Model Fit

R	R²
0.266	0.071

Model Coefficients

Predictor	Estimate	SE	t	p	Stand. Estimate
Intercept	2.577	0.155	16.63	< .001	—
age	−0.0254	0.0030	−8.46	< .001	−0.266

Saying the slope out loud

\[\widehat{transformers} = 2.58 - 0.025(age)\]

“Each additional year of age is associated with 0.025 fewer Transformers movies seen.”
That sounds like nothing. So rescale it to a unit people care about:
- 10 years → about 0.25 fewer movies
- 40 years → about one whole movie fewer
Note the word associated. Nobody was randomly assigned an age.

And the intercept?

$b_0 = 2.58$ = the predicted number of Transformers movies for a person aged zero.

Careful

Nobody in this sample is under 18. A prediction at age 0 is extrapolation — the model has never seen data anywhere near there.

The intercept is an anchor for the line, not a finding.

When zero is not meaningful, centering the predictor fixes the interpretation (we will do this for real in Week 9).

Predict a person, find a residual

By hand: $2.577 + (-0.0254 \times 20) = 2.07$ movies for a 20-year-old.

By JAMOVI: in Linear Regression → Save, tick Predicted values and Residuals. Two new columns appear in the Data tab — one row per person.

A 20-year-old in the data who has seen 4 movies:

\[e_i = Y_i - \hat{Y_i} = 4 - 2.07 = +1.93\]

Positive residual → above the line → the model under-predicted this person.

🛠️ Activity Stop 1: One Line, Two Stories

Tip

Part A: Follow along · ~20 min · Pairs

We fit this model together, from the top:

Plot it, then guess r before you run it
Fit the model; find the intercept and slope in the Coefficients table
Say the slope in a unit that means something
Predict a person and compute their residual

Open the activity

☕ Break

10 minutes

When we come back: how do we know if this line is any good?

Part 3: How Well Does the Line Fit?

Statistical inference

The way the world is = our model + error
How good is our model? Does it “fit” the data well?

To assess fit, we examine the proportion of variance in our outcome that is “explained” by our model.

To do that, we partition the variance into two categories: variability captured by the model, and variability that is not.

Partitioning variation

Start with the relationship between observed $Y$ and fitted $\hat{Y}$:

\[Y_i = \hat{Y}_i + e_i\]

Variances are additive when two variables are independent. If C = A + B, then the variance of C is the sum of the variances of A and B.

Why can we use that rule here?

Partitioning variation

$\hat{Y}_i$ and $e_i$ are independent of each other. So the variance of $Y$ equals the variance of $\hat{Y}$ plus the variance of $e$.

\[\large s^2_Y = s^2_{\hat{Y}} + s^2_{e}\]

Variances are sums of squares divided by N−1, and all have the same sample size here, so:

\[\large SS_Y = SS_{\hat{Y}} + SS_{\text{e}}\]

\[ SS_Y = SS_{\text{Model}} + SS_{\text{Residual}}\]

Note

Want to see those sums of squares? In Linear Regression, open Model Fit → Overall Model Test (F test). That F is $SS_{Model}$ and $SS_{Residual}$, each divided by its degrees of freedom.

Coefficient of Determination

\[\Large \frac{SS_{Model}}{SS_{Y}} = \frac{s_{Model}^2}{s_{y}^2} = R^2\]

$R^2$ is the proportion of variance in Y explained by the model

\[\large \sqrt{R^2} = r_{Y\hat{Y}}\]

With one predictor, that is just our old friend $r$.

Ta da! 🎉

Look back at the two tables you already have open:

Correlation Matrix

r = −0.266

(−0.266)² = 0.071

Linear Regression → Model Fit

R = 0.266

R² = 0.071

And the Standardized estimate for age? −0.266. The same number, wearing a third hat.

Significance $\neq$ importance

The $p$-value tells us the effect is likely not zero. It says nothing about whether the effect is large or meaningful.

Important

With a large enough sample, a tiny, trivial effect becomes statistically significant. Our n here is 938.

Statistical Significance $\neq$ Practical Significance

Effect size for a regression

The slope itself — in the units of your study. The most concrete thing you have.
Standardized estimate / $r$ — Cohen’s rough guide: small (|.10|), medium (|.30|), large (|.50|)
Coefficient of determination $R^2$ — proportion of variance accounted for

For our model: $r = -.27$, so $R^2 = .071$

“Approximately 7% of the variability in Transformers movies seen is accounted for by age.”

Writing it up

Template: b = slope, t(df) = statistic, p = p-value, plus $R^2$

Age significantly predicted the number of Transformers movies a respondent had seen, b = −0.03, t(936) = −8.46, p < .001. Each additional decade of age was associated with about a quarter of a movie fewer, and age accounted for 7.1% of the variance in movies seen ($R^2$ = .07).

Tip

Notice what the good version includes: the estimate in real units, the test, and a statement of how much it matters. All three.

JAMOVI’s tables are nearly APA-formatted already — right-click any table → Export (or copy) to drop it into Word.

Checking the residuals

Linear Regression > Assumption Checks

Residual Plots gives you residuals vs. fitted values, plus a Q-Q plot
Q-Q plot of residuals checks the normality of the errors

The residuals-vs-fitted plot should look like a shapeless cloud with no trend. Flat. By construction — and in Week 10 we will use that fact to catch models that are lying to us.

Part 4: The Lab 5 Punchline

A t-test is a regression

First the test you already know:

Analyses > T-Tests > Independent Samples T-Test

Dependent Variable: transformers · Grouping Variable: gender
Tick Student’s (equal variances) and Descriptives

(Filter to Male/Female first — Data → Filters → gender != "Other")

Test	statistic	df	p
Student’s t	−2.29	911	0.022

Group means: Female 1.21 · Male 1.44

The same question as a line

Analyses > Regression > Linear Regression

Dependent Variable: transformers
Factors: gender (categorical predictors go in Factors, not Covariates)

Predictor	Estimate	SE	t	p
Intercept	1.21	—	—	—
gender: Male – Female	0.23	0.10	2.29	0.022

The intercept (1.21) is the mean of the reference group (Female)
The estimate (0.23) is the difference between the means
Same t. Same p. One machine.

Who is the reference group?

JAMOVI picks the first level alphabetically by default — here, Female.

Linear Regression > Reference Levels

Change it to Male and watch what happens:

The intercept becomes 1.44 (the Male mean)
The estimate flips sign to −0.23
The t and p are unchanged

The model is the same model. You just asked it to answer from a different starting point.

Why this matters

Note

You do not have to memorize a separate procedure for every design.

Dependent Variable + Covariates/Factors handles the continuous case and the group case. Over the next few weeks we just add more predictors to the same boxes.

Next week: groups as regression, in full — t-tests and ANOVA as one model.

One more thing: regression to the mean

An observation about heights motivated the development of regression: if you selected an exceptionally tall (or short) parent, their child was almost always less extreme.

Regression to the mean: a random variable that produces an extreme score on a first measurement tends to produce a less extreme score on a second measurement.

Warning

This is a threat to internal validity whenever an intervention is assigned based on first-measurement scores.

Tutoring the students who bombed the midterm? They would have improved somewhat anyway.

🛠️ Activity Stop 2: Parts B & C

Tip

One Line, Two Stories · ~35 min · Pairs

Part B (~25 min): Do older people read more books?

You will get p = .027 — and then one person who reported reading 1,000 books will make it disappear
In JAMOVI that decision is a filter: Data → Filters → books <= 365
The p-value flips on a judgment call you make

Part C (~10 min): The t-test / regression equivalence, in your own file

Open the activity

JAMOVI ↔︎ R cheat sheet

What you want	JAMOVI	R
Descriptives	Exploration → Descriptives	`describe()` / `summary()`
Scatterplot + line	scatr → Scatterplot, Linear	`ggplot() + geom_smooth(method = "lm")`
Correlation	Regression → Correlation Matrix	`cor.test()`
Fit a line	Regression → Linear Regression	`lm(y ~ x, data = d)`
Continuous predictor	Covariates box	just add it
Categorical predictor	Factors box	`factor()`
Standardized slope	Model Coefficients → Standardized estimate	`lm.beta()` / scale the variables
R²	Model Fit → R²	`summary(fit)$r.squared`
Predicted values & residuals	Save → Predicted values, Residuals	`broom::augment(fit)`
Drop cases	Data → Filters	`filter()`
Change reference group	Reference Levels	`relevel()` / `fct_relevel()`

✍️ Wrap-Up

Finish these at the bottom of your activity file:

A slope, stated in the units of a study, without using the word “slope.”
One intercept from today that is meaningless, and why.
One thing that is still muddy → put it in your journal this week.

For next time

📋 Lab 6 — Reading the Line, due Tuesday 9/29 at 11:59pm

Almost no coding — and nothing that requires R or JAMOVI. I give you the output; you tell me what it means.
Uses the Lab 5 data plus a new context

📖 Chapter 15 — ST

Schedule change

The lab schedule has moved up from the PDF syllabus. Lab 6 is due 9/29, the midterm is assigned 10/15 and due 10/27. Website = current version.

Week 05: Simple Regression with JAMOVI

Today…

First, a confession about Lab 5

Why we teach regression first

Uses for regression

Part 1: Building the Line

The core idea

A line you already know

Let’s “science up” the equation

Expected vs. Actual

OLS

OLS

🔍 Watch OLS happen

Where does the slope come from? 🤔

Regression coefficient, \(b_{1}\)

Slope is the correlation (rescaled)

Estimating the intercept, \(b_0\)

Part 2: Reading a Line Out Loud

Our data today 🃏

Step 1: ALWAYS visualize first

Step 2: Descriptives first

Step 3: The correlation

Step 4: The same relationship as a line

Saying the slope out loud

And the intercept?

Predict a person, find a residual

🛠️ Activity Stop 1: One Line, Two Stories

☕ Break

Part 3: How Well Does the Line Fit?

Statistical inference

Partitioning variation

Partitioning variation

Coefficient of Determination

Ta da! 🎉

Significance \(\neq\) importance

Effect size for a regression

Writing it up

Checking the residuals

Part 4: The Lab 5 Punchline

A t-test is a regression

The same question as a line

Who is the reference group?

Why this matters

One more thing: regression to the mean

🛠️ Activity Stop 2: Parts B & C

JAMOVI ↔︎ R cheat sheet

✍️ Wrap-Up

For next time