Regression refresher,
intro to group-structured data


Data Analysis for Psychology in R 3

Elizabeth Pankratz (elizabeth.pankratz@ed.ac.uk)


Department of Psychology
University of Edinburgh
2026–2027

Course Overview


Linear mixed models
(with Dr. Elizabeth Pankratz)		 Regression refresher, intro to group-structured data
Modelling group-structured data using random effects
Interpreting LMMs and building maximal models
Troubleshooting model fit, checking assumptions + diagnostics
LMMs: Practice analysis
factor analysis
working with multi-item measures
(with Dr. Josiah King)		 measurement and dimensionality
exploring underlying constructs (EFA)
testing theoretical models (CFA)
reliability and validity
recap & exam prep

Warm-up

Part 1: Brain dump all the stats terms you know/remember


Here are some visuals that might help jog your memory:

Part 2: Connect key ideas

Example terms:

  • standard deviation (SD)
  • standard error (SE)
  • confidence interval (CI)

Individually or with your seatmates:

  1. Relax while Elizabeth transfers key terms to the hexagon sheet
  2. Using a big-screened device, access hexagons at this link: https://edin.ac/4cQpsUM
  3. Duplicate Slide 1 and work on your own duplicated copy
  4. Click and drag each term from the menu onto its own hexagon. If two hexagons are touching, then the ideas on each hexagon are somehow connected.

There are no wrong answers here! The purpose of this activity is to help you remember a few of the ways that a few big ideas fit together.

Part 3: Think about the properties of lines


Linear regression refresher

Data: After one week of mindfulness treatments, life satisfaction ratings from 36 people

Code 
set.seed(1)
p_lifesat <- lifesat |>
  ggplot(aes(x = condition, y = lifesat, fill = condition, colour = condition)) +
  geom_violin(alpha = 0.5) +
  geom_jitter(alpha = 0.5, size = 5) +
  stat_summary(geom = 'point', fun = mean, colour = 'black', size = 8) +
  theme(legend.position = 'none') +
  scale_colour_manual(values = pal) +
  scale_fill_manual(values = pal) +
  NULL
p_lifesat

lifesat |>
  head(15)
# A tibble: 15 x 3
   condition ppt_id lifesat
   <chr>     <chr>    <dbl>
 1 journal   j1        34.2
 2 journal   j2        42.5
 3 journal   j3        45.6
 4 journal   j4        44.6
 5 journal   j5        35.7
 6 journal   j6        50.2
 7 journal   j7        31.9
 8 journal   j8        40.7
 9 journal   j9        57.2
10 journal   j10       37.0
11 journal   j11       33.0
12 journal   j12       46.9
13 journal   j13       34.7
14 journal   j14       36.1
15 journal   j15       60.2

Numerically, there is a difference between condition means: “meditate” has higher lifesat than “journal”.

We want to test: Is that difference between condition means sufficiently unlikely to equal zero?

Is the difference between condition means sufficiently unlikely to equal zero?

A linear model will fit a line that passes through the two group means.

  • To answer our question, we will first imagine that in reality, the true slope of this line is zero: the groups are not different at all (= our null hypothesis).

  • In that reality where the groups are not different, we know the distribution of values that the slope could reasonably have.

  • We compare that null distribution to the slope we have actually observed.

  • If our observed slope falls in either tail of the null distribution, then we can say that the slope is significantly different from zero!

Fit the model

lifesat_mod <- lm(
  lifesat ~ condition,   # outcome ~ predictor
  data = lifesat
)
summary(lifesat_mod)

Call:
lm(formula = lifesat ~ condition, data = lifesat)

Residuals:
    Min      1Q  Median      3Q     Max 
-25.467  -4.980  -0.274   5.933  19.280 

Coefficients:
                  Estimate Std. Error t value   Pr(>|t|)    
(Intercept)          40.95       2.17   18.88    < 2e-16 ***
conditionmeditate    18.47       3.07    6.02 0.00000081 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 9.2 on 34 degrees of freedom
Multiple R-squared:  0.516, Adjusted R-squared:  0.502 
F-statistic: 36.2 on 1 and 34 DF,  p-value: 0.000000812

Interpret model coefficients

Coefficients:
                   Estimate Std. Error t value   Pr(>|t|)    
 (Intercept)          40.95       2.17   18.88    < 2e-16 ***
 conditionmeditate    18.47       3.07    6.02 0.00000081 ***

(Intercept) = the estimated average lifesat for condition = 0, the reference level, which is journal.

  • People who journal are estimated to have a life satisfaction of 40.95 points, on average.
  • This estimate is significantly different from zero (\(p\) < .001), but that’s not very surprising or interesting.

conditionmeditate: = the difference between the average lifesat when condition = 1 = meditate and the average lifesat when condition = 0 = journal.

  • People who meditate are estimated to have a life satisfaction that’s 18.47 points greater than people who journal.
  • This estimate is significantly different from zero (\(p\) < .001).

So yes, the difference between condition means is sufficiently unlikely to equal zero!

We reject the H0 and declare that the groups are significantly different.

Now that we’re warmed up: welcome to DAPR3 ✨

This week’s learning objectives


What is a grouping variable?

Why are grouping variables important to know about?

What does a statistical model aim to represent / to capture / to formalise?

What are the different kinds of variability that a statistical model must account for?

Between-subjects vs. 
repeated measures

Between-subjects vs. repeated measures: What do these terms mean to you?


An example of repeated-measures data:
One week of mindfulness treatments

Code 
p_lifesat_week <- lifesat_week |>
  ggplot(aes(x = day, y = lifesat, colour = condition)) +
  geom_smooth(method ='lm', se = F, formula = 'y ~ x', linewidth = 2) +
  geom_jitter(alpha = 0.5, width = .2, size = 5) +
  scale_colour_manual(values = pal) +
  scale_x_continuous(breaks = 1:7) +
  theme(
    legend.position = 'bottom',
    panel.grid.minor.x = element_blank()
  ) +
  NULL
p_lifesat_week

lifesat_week |>
  head(15)
# A tibble: 15 x 4
     day condition ppt_id lifesat
   <dbl> <chr>     <chr>    <dbl>
 1     1 journal   j1        36.6
 2     2 journal   j1        33.8
 3     3 journal   j1        44.2
 4     4 journal   j1        25.3
 5     5 journal   j1        42.4
 6     6 journal   j1        45.2
 7     7 journal   j1        34.2
 8     1 journal   j2        22.5
 9     2 journal   j2        40.5
10     3 journal   j2        34.7
11     4 journal   j2        43.5
12     5 journal   j2        41.1
13     6 journal   j2        44.4
14     7 journal   j2        42.5
15     1 journal   j3        51.8

(In the example from before, we were looking at data from Day 7 only!)

RQs: Does life satisfaction increase, the longer people take part in treatment? Is one treatment associated with greater life satisfaction than the other?

How do we know that this is
repeated measures data?

lifesat_week |>
  head(12)
# A tibble: 12 x 4
     day condition ppt_id lifesat
   <dbl> <chr>     <chr>    <dbl>
 1     1 journal   j1        36.6
 2     2 journal   j1        33.8
 3     3 journal   j1        44.2
 4     4 journal   j1        25.3
 5     5 journal   j1        42.4
 6     6 journal   j1        45.2
 7     7 journal   j1        34.2
 8     1 journal   j2        22.5
 9     2 journal   j2        40.5
10     3 journal   j2        34.7
11     4 journal   j2        43.5
12     5 journal   j2        41.1


  • We have data from multiple ppt_ids (j1, j2, etc.), and each of those ppt_ids appears more than once in our dataset.

  • Specifically, each participant has one lifesat value per day.

  • In other words, each participant has contributed repeated measurements to the dataset.


Because each participant contributed multiple observations, we call ppt_id a “grouping variable”.

  • day and condition also contain multiple observations of the same value. In this way, they are also grouping variables. But they’re conceptually a bit different from ppt_id, as we’ll see soon.

Why are grouping variables important?

Why are grouping variables important?


If you took DAPR2, you might remember that linear models make four assumptions:

  1. Linearity of association: The association between predictors and outcome can reasonably be modelled as a straight line.
  2. Independence of errors: Each data point is independent from each other data point. (Knowing something about one data point tells you nothing about any of the other data points.)
  3. Normality of errors: The model’s errors (the distance between observed point and model-estimated line) are normally-distributed.
  4. Equal variance of errors: The model’s errors have similar variance across the whole range of model-fitted values.


As soon as we have repeated-measures data with grouping variables like ppt_id, the data points are no longer independent.

Why aren’t the data points independent?

Code 
lifesat_week |>
  filter(ppt_id %in% c('j16', 'm9')) |>
  ggplot(aes(x = day, y = lifesat, colour = ppt_id, shape = ppt_id)) +
  geom_point(size = 7) +
  scale_colour_manual(values = pal[2:3]) +
  scale_x_continuous(breaks = 1:7) +
  theme(
    legend.position = 'bottom',
    panel.grid.minor.x = element_blank()
  ) +
  NULL

If we know one data point from m9, we can make some good guesses about what values m9’s other data points might have. Same for j16.

  • In other words: Some of our data gives us information about other parts of our data.
  • In other words: Data points from one person are not independent of other data points from that same person.

We use grouping variables to tell the model that data points are not independent, and instead are grouped together in this way.

  • The specifics of how we tell the model about grouping variables: next week.
  • This week: how we identify grouping variables in any dataset.

How to identify grouping variables: Two examples

Ex. 1: Mindfulness treatments, Day 7

What categorical variables do we have in lifesat?

glimpse(lifesat)
Rows: 36
Columns: 3
$ condition <chr> "journal", "journal", "journal", "journal", "journal", "jour~
$ ppt_id    <chr> "j1", "j2", "j3", "j4", "j5", "j6", "j7", "j8", "j9", "j10",~
$ lifesat   <dbl> 34.2, 42.5, 45.6, 44.6, 35.7, 50.2, 31.9, 40.7, 57.2, 37.0, ~

Ex. 1: Grouping and counting

For condition:

lifesat |>
  group_by(condition) |>
  count()
# A tibble: 2 x 2
# Groups:   condition [2]
  condition     n
  <chr>     <int>
1 journal      18
2 meditate     18

For ppt_id:

lifesat |>
  group_by(ppt_id) |>
  count()
# A tibble: 36 x 2
# Groups:   ppt_id [36]
   ppt_id     n
   <chr>  <int>
 1 j1         1
 2 j10        1
 3 j11        1
 4 j12        1
 5 j13        1
 6 j14        1
 7 j15        1
 8 j16        1
 9 j17        1
10 j18        1
# i 26 more rows

Each value of condition appears more than once.

condition is a grouping variable   ✅

Each value of ppt_id appears only once.

ppt_id is not a grouping variable   ❌

Ex. 2: A week of mindfulness treatments

What categorical variables do we have in lifesat_week?

glimpse(lifesat_week)
Rows: 252
Columns: 4
$ day       <dbl> 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, ~
$ condition <chr> "journal", "journal", "journal", "journal", "journal", "jour~
$ ppt_id    <chr> "j1", "j1", "j1", "j1", "j1", "j1", "j1", "j2", "j2", "j2", ~
$ lifesat   <dbl> 36.6, 33.8, 44.2, 25.3, 42.4, 45.2, 34.2, 22.5, 40.5, 34.7, ~

Ex. 2: Grouping and counting

For day:

lifesat_week |>
  group_by(day) |>
  count()
# A tibble: 7 x 2
# Groups:   day [7]
    day     n
  <dbl> <int>
1     1    36
2     2    36
3     3    36
4     4    36
5     5    36
6     6    36
7     7    36

For condition:

lifesat_week |>
  group_by(condition) |>
  count()
# A tibble: 2 x 2
# Groups:   condition [2]
  condition     n
  <chr>     <int>
1 journal     126
2 meditate    126

For ppt_id:

lifesat_week |>
  group_by(ppt_id) |>
  count()
# A tibble: 36 x 2
# Groups:   ppt_id [36]
   ppt_id     n
   <chr>  <int>
 1 j1         7
 2 j10        7
 3 j11        7
 4 j12        7
 5 j13        7
 6 j14        7
 7 j15        7
 8 j16        7
 9 j17        7
10 j18        7
# i 26 more rows

Each value of day appears more than once.

day is a grouping variable   ✅

Each value of condition appears more than once.

condition is a grouping variable   ✅

Each value of ppt_id appears more than once.

ppt_id is a grouping variable   ✅

Ex. 2: Conceptual differences between these grouping variables


Our RQs are about how different conditions and different days are associated with lifesat.


  • Notice that these RQs refers to only two of the grouping variables, condition and day.
  • condition and day are manipulated / controlled by our experimental design.
  • The RQ does not refer to the third grouping variable, ppt_id.
  • ppt_id is not manipulated / controlled by the experimental design.
  • But our statistical model needs to account for it anyway.
  • The first block of DAPR3 is all about learning how to do that.


Thinking in models:
What process generated the outcome data we observe?

Ex. 1: Mindfulness treatments, Day 7


Ex. 2: A week of mindfulness treatments

When we have multiple observations from a single source (here, a single person), we have something new: random, non-manipulated variability that comes from a specific variable.

Identifying reproducible vs. random variability: Strategy 1

Ask yourself: If the same RQ were tested again in a different experiment, which variables would have to stay the same, and which variables could potentially be different?


Does the variable need to contain the same values, or else the RQ could not be tested?

  • If so: this variable contributes manipulated / controlled / reproducible variability.
  • e.g.:
    • experimental manipulations
    • covariates to control for other influences (e.g., handedness, age, gender)


Could the variable potentially contain different values, and the RQ would still be testable?

  • If so: this variable contributes non-manipulated / non-controlled / random variability.
  • e.g.:
    • participants/subjects
    • stimuli

Identifying reproducible vs. random variability: Strategy 2

Our goal is often to generalise the results of our analysis to a broader population.
Ask yourself: What variables can and cannot be generalised over?

If your analysis should be able to generalise across all potential levels of a grouping variable, even beyond the ones you observed, then the grouping variable contributes non-manipulated / non-controlled / random variability.

For example: Do we want our results to generalise to all possible experimental conditions, beyond the ones we studied?

  • The question doesn’t really make sense … what other experimental conditions even are there?
  • We only want to make statements about the specific experimental conditions we’re testing—that’s why we’re testing them.
  • So a variable that represents experimental conditions does not contribute random variability to the data.

But: Do we want our results to generalise to all possible people, beyond the ones we studied?

  • Yeah, that would be ideal!
  • So a variable that represents different people does contribute random variability to the data.

Back matter

Learning objectives revisited

What is a grouping variable?

  • A categorical variable whose values appear more than once in the dataset.

Why are grouping variables important to know about?

  • Because if we have multiple observations from a single source (e.g., a single person) which contribute random variability to our data, then our observations are no longer independent.
  • Therefore the linear model’s assumption about independent errors is not met.

What does a statistical model aim to represent / to capture / to formalise?

  • The process that generated the data we are modelling.

What are the different kinds of variability that a statistical model must account for?

  • Manipulated / controlled / reproducible variability comes from from our predictor variables.
  • Non-manipulated / non-controlled / random variability comes from our grouping variables.
  • Other random variability not associated with specific variables is modelled as residual error.

To do this week


Tasks:


Work on exercises in labs


Complete the weekly quiz

Get support:


Consult the flash cards


Ask questions anonymously on Piazza


We really like seeing you in office hours!

Appendix

Line plots from Wooclap

Code 
p1 <- ggplot() +
  geom_abline(aes(intercept = 1, slope = 1), linewidth = 2) +
  scale_x_continuous(limits = c(-2, 2), breaks = -2:2) +
  scale_y_continuous(limits = c(0, 5), breaks = 0:5) +
  theme(panel.grid.minor = element_blank()) +
  labs(x = 'x', y = 'y') +
  NULL
# ggsave('figs/woo/w1-p1.png', width = 10, height = 10, units = 'cm')

p2 <- ggplot() +
  geom_abline(aes(intercept = 0, slope = 1), linewidth = 2) +
  scale_x_continuous(limits = c(0, 2), breaks = 0:2) +
  scale_y_continuous(limits = c(0, 3), breaks = 0:3) +
  theme(panel.grid.minor = element_blank()) +
  labs(x = 'x', y = 'y') +
  NULL
# ggsave('figs/woo/w1-p2.png', width = 10, height = 10, units = 'cm')

p3 <- ggplot() +
  geom_abline(aes(intercept = 0, slope = -10), linewidth = 2) +
  scale_x_continuous(limits = c(-2, 2)) +
  scale_y_continuous(limits = c(-30, 30), breaks = seq(-30, 30, by=10)) +
  theme(panel.grid.minor = element_blank()) +
  labs(x = 'x', y = 'y') +
  NULL
# ggsave('figs/woo/w1-p3.png', width = 10, height = 10, units = 'cm')

p4 <- ggplot() +
  geom_abline(aes(intercept = 1, slope = -1), linewidth = 2) +
  scale_x_continuous(limits = c(0, 4), breaks = 0:4) +
  scale_y_continuous(limits = c(-3, 1), breaks = -3:5) +
  theme(panel.grid.minor = element_blank()) +
  labs(x = 'x', y = 'y') +
  NULL
# ggsave('figs/woo/w1-p4.png', width = 10, height = 10, units = 'cm')

p1 + p2 + p3 + p4