Week 08: Multiple Regression I

JAMOVI version · Date: October 15, 2026

Today…

One predictor was never the plan.

• Why a second predictor changes the first one’s meaning
• “Holding constant” — what it actually does
• A slope that flips sign when we add the right variable
• Did the new predictor earn its place? (Model Builder + Model Comparisons)
• b or β: which one do you report?
• When predictors step on each other (collinearity)

Note

This is the JAMOVI version of Week 8. Same models, same numbers, same interpretations — all point-and-click. The R version is here.

Where we have been

Week	Model	What was new
5	\(Y = b_0 + b_1X\)	one continuous predictor (Covariates)
6	\(Y = b_0 + b_1D\)	the predictor can be a group (Factors)
7	\(SS_{total} = SS_{model} + SS_{res}\)	how well the line fits

Everything so far has been one predictor at a time. That is almost never the model you want.

The only equation we need

\[Y_i = b_0 + b_1X_{1i} + b_2X_{2i} + \dots + b_kX_{ki} + e_i\]

• \(b_0\) — predicted \(Y\) when every predictor is 0
• \(b_1\) — change in \(Y\) per 1-unit change in \(X_1\), holding the other predictors constant
• \(e_i\) — still the part we miss

Important

In JAMOVI you build this model by dragging more than one variable into Covariates. That is the entire mechanical change. Everything else today is about what the numbers now mean.

Part 1: Why Bother?

You have been hired by the county 🚒

The fire department wants to know what drives the cost of a fire.

300 fires: how many trucks were dispatched, how many acres burned, and the damage in thousands of dollars.

🖱️ Open the data

☰ Menu → Open → Browse → fire_data.csv

Four columns: fire_id, acres, trucks, damage. Confirm all three analysis variables are Continuous.

Start where we always start

🖱️ Look at it first

Exploration → scatr → Scatterplot

• trucks → X-Axis
• damage → Y-Axis
• Regression Line: Linear

A clear, tight, positive cloud. More trucks, more damage.

The bivariate model

🖱️ Fit it

Regression → Linear Regression

• damage → Dependent Variable
• trucks → Covariates
• Model Coefficients: ✅ Estimate, ✅ Standard error, ✅ p-value
• Model Fit: ✅ R², ✅ Adjusted R², ✅ Overall model test (F test)

Predictor	b	SE	t	p
Intercept	8.84	3.28	2.70	.007
trucks	12.73	0.51	24.73	< .001

R² = .672

What would you tell the county?

• b = 12.73: each additional truck is associated with $12,730 more damage
• \(p\) < .001, \(R^2\) = .67 — this is not a fragile finding

“Trucks strongly predict damage. Send fewer trucks.”

Warning

Every number in that sentence is correct. The recommendation would get people killed.

🧠 Side quest: draw the thing

Before you fit anything, sketch what you think is going on. Arrows for “causes.”

• Does trucks → damage?
• Does something cause both?

A variable that causes both your predictor and your outcome is a confound. Leave it out and your slope absorbs its effect.

These sketches have a name — Directed Acyclic Graphs — and Dr. Haraden is about to abuse the chalkboard with one. 🧑‍🏫

The variable we ignored

🖱️ Colour the cloud by fire size

Exploration → scatr → Scatterplot

• trucks → X-Axis, damage → Y-Axis
• acres → Group (JAMOVI will bin a continuous grouping variable; alternatively make a 3-level size variable under Data → Compute using IF)

Big fires get more trucks and do more damage. The slope we estimated was mostly fire size wearing a truck costume.

Put acres in the model

🖱️ One extra drag

In the same Linear Regression panel, drag acres into Covariates underneath trucks.

Predictor	b	SE	t	p
Intercept	10.94	1.65	6.62	< .001
trucks	−2.63	0.58	−4.54	< .001
acres	4.09	0.14	29.64	< .001

R² = .917, adj. R² = .917

The sign flipped

• trucks: +12.73 → −2.63
• acres: +4.09 per acre
• \(R^2\): .672 → .917

Each additional truck is associated with $2,630 less damage — among fires of the same size.

What just happened

Before

We compared a 2-truck fire to a 9-truck fire. Those are not the same fire. One was a kitchen; one was a neighborhood.

After

We compare 2-truck and 3-truck fires of the same acreage. Now the extra truck can only help.

“Holding constant” is a comparison, not a magic word

\(b_1\) answers: among cases that are identical on every other predictor in the model, what is the difference in \(Y\) per unit of \(X_1\)?

Identical on the predictors you included. Nothing else.

Omitted variables do not politely disappear

When you leave out a variable that is related to both \(X\) and \(Y\), its effect does not vanish — it gets folded into the coefficients you did estimate.

• Sometimes the slope is inflated (the trucks case)
• Sometimes it is shrunk (next week’s tutoring example)
• Sometimes it appears out of nowhere (Lab 8’s suppression)
• Sometimes nothing moves — and that is a finding too

You cannot tell which from the output. You can only tell from what you chose to measure.

Part 2: Real Data, Real Stakes

520 students, one survey 😵‍💫

A student-stress survey: everything rated 1–5.

🖱️ Open and rename

☰ Menu → Open → Browse → student_stress.xlsx

The column headers are full sentences (with emoji). Double-click each header and give it a short Name:

Original question	Name it
Rate your sleep quality	sleep_quality
Headaches per week	headaches
Rate your academic performance	academic_perf
Rate your study load	study_load
Extracurriculars per week	extracurricular
Rate your stress levels	stress

Set all six to Continuous.

Describe, then correlate

🖱️ Two panels

1. Exploration → Descriptives — all six variables, ✅ Mean, ✅ SD, ✅ Minimum, ✅ Maximum
2. Regression → Correlation Matrix — all six variables

	r with stress
study_load	.392
sleep_quality	.165
extracurricular	.052
academic_perf	.055
headaches	−.072

And the two predictors we care about barely know each other: r(study_load, sleep_quality) = .065

Model 1: study load alone

🖱️ Linear Regression

stress → Dependent Variable · study_load → Covariates

Predictor	b	SE	t	p
Intercept	1.810	0.123	14.72	< .001
study_load	0.388	0.040	9.69	< .001

R² = .153, adj. R² = .152, RSE = 1.251

Each 1-point increase in study load → 0.39 points more stress.

Model 2: add sleep quality

🖱️ Drag sleep_quality into Covariates

Predictor	b	SE	t	p
Intercept	1.292	0.191	6.76	< .001
study_load	0.378	0.040	9.54	< .001
sleep_quality	0.174	0.050	3.50	< .001

Predictor	Alone	Together
study_load	0.388	0.378

The study-load slope moved by 0.01. Essentially nothing.

Why nothing moved

A coefficient only changes when the new predictor is related to both the outcome and the old predictor.

• r(study_load, sleep_quality) = .065 → the two predictors carry separate information
• So each one keeps the effect it had on its own
• Their contributions simply add

Tip

“I added a predictor and the first one didn’t budge” is a real result, and it is the common one. It means your two predictors are not competing for the same variance. Say that, instead of apologizing for it.

Interpreting Model 2 out loud

\[\widehat{Stress} = 1.29 + 0.378(StudyLoad) + 0.174(SleepQuality)\]

• 1.29 — predicted stress for a student who rated study load and sleep quality as 0. Neither is possible (the scale starts at 1), so this intercept is a mathematical anchor, not a person.
• 0.378 — comparing two students with the same sleep quality, one point more study load goes with 0.38 points more stress
• 0.174 — comparing two students with the same study load, one point more sleep quality goes with 0.17 points more stress

Wait. Read that last one again. 🤨

Better sleep quality predicts more stress (b = +0.174, p < .001), holding study load constant.

Three possibilities, and you cannot choose between them from this output:

1. The item is scored backwards — maybe 5 means “I rate my sleep quality as a problem”
2. The relationship is real and reflects something else (students who sleep well take on more)
3. The survey question was ambiguous and students interpreted it inconsistently

Important

This is the midterm’s lesson arriving in a lecture. Check the direction of every coefficient against what you expected. When it disagrees, the first suspect is always scoring — not the universe.

Part 3: Did It Earn Its Place?

JAMOVI does this better than R 🏆

🖱️ Model Builder

Inside Linear Regression, open the Model Builder section:

1. study_load → Block 1
2. Click + Add New Block
3. sleep_quality → Block 2

Then open Model Comparisons and tick Compare models.

JAMOVI now reports both models side by side and the test of the difference. In R this takes a second lm() and an anova() call.

The Model Fit Comparison table

Model	R²	adj. R²	RSE
1. study load	.1534	.1517	1.2506
2. + sleep quality	.1730	.1698	1.2372

Model Comparisons: ΔR² = .0196, F(1, 517) = 12.27, p < .001

• About 2% more of stress accounted for
• Adjusted \(R^2\) went up, so the predictor paid its own df
• Typical miss shrank from 1.251 to 1.237 (on a variable with SD = 1.358)

What that F actually is

The comparison test asks: does the drop in unexplained variability exceed what one extra predictor would buy by chance?

• Residual SS: 810.12 → 791.34
• The new predictor picked up 18.78 of it
• \(F\)(1, 517) = 12.27, p = .0005

With one added predictor, this \(F\) is just the square of its \(t\): \(3.503^2 = 12.27\). Same test, different table. ✅

Now let’s add three useless things

🖱️ Block 3

Add extracurricular, headaches and academic_perf together as Block 3.

Predictor	b	p
study_load	0.383	< .001
sleep_quality	0.178	< .001
extracurricular	−0.015	.697
headaches	−0.074	.097
academic_perf	−0.038	.483

Watch the two R²s diverge

Model	R²	adj. R²	RSE
2 predictors	.1730	.1698	1.2372
5 predictors	.1779 ⬆️	.1699 ➡️	1.2371

• \(R^2\) rose by .005, because it always rises
• Adjusted \(R^2\) moved by .0001 — three predictors bought you nothing
• RSE: unchanged to four decimals

Important

Week 7’s warning, now with consequences. \(R^2\) cannot tell you whether a predictor belongs in the model. Adjusted \(R^2\), the model comparison, and your theory can.

Part 4: Reporting It

b or β?

🖱️ Standardized estimates

Model Coefficients → ✅ Standardized estimate

JAMOVI adds a Stand. Estimate column. (It standardizes continuous predictors only — which is what you want.)

Report b when

• the units mean something (dollars, hours, mg)
• you want to compare predictors within this sample

Report β when

• the units are arbitrary (survey points)
• you want to compare across studies or scales

Here: β = .38 (study load) vs .14 (sleep quality). Study load matters roughly 2.7× as much.

When predictors step on each other

If two predictors are nearly the same variable, the model cannot tell them apart. Coefficients get unstable and standard errors balloon.

🖱️ Check it

Assumption Checks → ✅ Collinearity statistics

JAMOVI prints VIF and tolerance for each predictor.

• Stress model: VIF = 1.004 — these predictors are strangers
• Fire model: VIF = 5.00 — acres and trucks are nearly redundant (r = .89), and the SE on trucks rose from 0.515 to 0.579 as a result

Rules of thumb: > 5 is worth a sentence, > 10 is a problem. Note that the fire model still worked — high VIF widens your SEs, it does not invalidate the model.

The write-up

Stress was regressed on self-reported study load and sleep quality in a sample of 520 students. The overall model was significant, F(2, 517) = 54.07, p < .001, accounting for 17.3% of the variance in stress (adj. R² = .170). Holding sleep quality constant, each one-point increase in study load was associated with a 0.38-point increase in stress (b = 0.38, SE = 0.04, t = 9.54, p < .001). Holding study load constant, each one-point increase in rated sleep quality was associated with a 0.17-point increase in stress (b = 0.17, SE = 0.05, t = 3.50, p < .001). Adding sleep quality improved fit over study load alone, ΔR² = .020, F(1, 517) = 12.27, p < .001. The positive coefficient for sleep quality was contrary to expectation and may reflect the direction in which that item was scored.

Tip

Everything a reader needs: the overall test, each coefficient with its holding-constant clause, the model comparison, and an honest flag on the odd result.

🛠️ Activity: Does Spending More Make More?

Tip

Week 8 Exercises · ~45 min · Pairs

1,763 films, 1970–2013: budget, international gross, year, and whether the film passes the Bechdel test.

• Budget alone explains 35% of gross
• Add one more variable and the budget slope gets bigger
• You will find out why, and write it up like a consultant

Every step in the activity is given in both R and JAMOVI.

Open the activity

✍️ Wrap-Up

Finish these at the bottom of your activity file:

1. In your own words: what does “holding X constant” let you compare?
2. A colleague says “my \(R^2\) went up, so the new predictor helped.” What do you ask them for?
3. One thing that is still muddy → journal.

For next time

📂 Midterm Project — due Tuesday 10/27 at 11:59pm

Part 6 is now unlocked. You have everything you need: the two-predictor model, the “holding constant” interpretation, ΔR² via Model Builder → Model Comparisons, and the model-comparison table.

Target for this weekend: Parts 1–5 done, Part 6 drafted. That leaves next week for writing instead of panicking.

No lab is due this week or next — the midterm is the assignment.

📖 Chapter 14.3 — ST · Chapter 8.1 — MSR

Next week: what happens when the groups from Week 6 meet the predictors from this week — and what to do when a slope is different for different groups.