Tutorial 5 (Advanced): Linking Brain to Phenotypes (Part 2: Correlation/Regression)

Correlation: Does Age Correlate With DMN connectivity¶

Overview¶

In the previous tutorial on Functional Connectivity, we learned that functional connectivity is commonly estimated using Pearson correlation between the fMRI time series of two brain regions.

For example, we calculated the correlation between the activity of Region A and Region B to obtain a single functional connectivity value.

Now, we will use the same statistical idea in a different way.

Instead of asking:

Are the time series of two brain regions correlated?

we now ask:

Is a brain connectivity measure correlated with age?

In other words, we are no longer correlating brain with brain, but rather brain with behavior.

Correlation measures the strength and direction of a relationship between two variables.

As a reminder:

• Correlation = +1 indicates a perfect positive relationship. As one variable increases, the other always increases.

• Correlation = 0 indicates no consistent relationship between the variables.

• Correlation = −1 indicates a perfect negative relationship. As one variable increases, the other always decreases.

Although different correlation methods produce slightly different statistics (for example, Pearson’s r, Spearman’s ρ, or Kendall’s τ), they all aim to answer the same general question:

Do two variables tend to vary together?

Resting-State Networks: DMN¶

So far, we have examined whether overall functional connectivity differs with age.

However, we haven’t yet explored whether age-related changes is associated with any specific functional networks or connectivity!

To narrow our scope, let’s look at one network that has received considerable attention in the neuroscience literature: the Default Mode Network (DMN). The DMN is involved in internally directed cognitive processes such as autobiographical memory, mind-wandering, self-referential thinking, and future planning. Numerous studies have shown that the DMN undergoes functional changes during healthy aging and in several neurological and psychiatric disorders.

For this reason, we will focus on connectivity involving the DMN. Specifically, we will examine nine functional connectivity measures:

• overall within-network functional connectivity,

• overall between-network functional connectivity,

• within-DMN connectivity,

• DMN–Visual connectivity,

• DMN–SMN connectivity,

• DMN–DAN connectivity,

• DMN–VAN connectivity,

• DMN–Limbic connectivity,

• DMN–FPN connectivity.

Parametric vs Non-Parametric Correlation Measures¶

There are several different ways to measure correlation, and the appropriate method depends on the type of data being analyzed.

In the previous tutorial, we calculated Pearson’s correlation coefficient (r) between the fMRI time series of two brain regions to estimate functional connectivity. Pearson’s correlation is a parametric correlation measure, meaning that it makes assumptions about the data. For example, it assumes that the variables are continuous and that their relationship is approximately linear.

However, Pearson’s correlation is not always the most appropriate choice.

More broadly, correlation measures can be divided into two categories:

• Parametric correlation measures, such as Pearson’s correlation (r), which make assumptions about the underlying data.

• Non-parametric correlation measures, such as Kendall’s rank correlation (τ) and Spearman’s rank correlation (ρ), which make fewer assumptions and instead use the rank, or ordering, of the observations.

For this tutorial, we will use non-parametric Kendall’s rank correlation coefficient (τ).

Why?

The LEMON dataset stores age in ordered age bins (for example, 20–25 years, 25–30 years, 30–35 years, and so on) rather than each participant’s exact age. This means that participants aged 20 and 24 are placed into exactly the same category, even though they are not the same age. Similarly, participants aged 25 and 29 are grouped together in the next category.

Because the data are organized into ordered groups rather than exact numerical values, it is more appropriate to treat age as an ordinal variable. In other words, we know the order of the age groups (younger to older), but we do not use the precise age of each participant.

Kendall’s rank correlation is designed for exactly this type of data. Kendall’s rank correlation does not care about the exact numerical values. Instead, it asks whether the ordering of one variable agrees with the ordering of the other variable.

Because our age variable is ordinal (ordered categories rather than exact numerical values), Kendall’s rank correlation is a natural and robust choice for this analysis.

Finally, because we are testing nine connectivity measures, we increase our chances of finding statistically significant results purely by chance. To account for this, we will correct for multiple comparisons using both Bonferroni correction and the Benjamini–Hochberg False Discovery Rate (FDR) procedure.

Computing Kendall Rank Correlation¶

For each FC measure, we will compute:

• the Kendall tau correlation coefficient

• the raw p-value

• the Bonferroni-adjusted p-value

• the Benjamini–Hochberg (FDR) adjusted p-value

A positive tau means the FC measure tends to be higher in the older group. A negative tau means it tends to be higher in the younger group.

Linear Regression: Age and Functional Connectivity¶

In the previous section, we used correlation to ask a simple question:

Is functional connectivity associated with age?

Correlation is an excellent tool for identifying whether two variables tend to vary together. However, it has some limitations. For example, it does not estimate how much functional connectivity changes with age, nor can it easily be extended to include additional variables such as sex, education, or depression scores.

To answer these kinds of questions, we use linear regression.

Linear regression fits a straight line that best describes the relationship between a predictor variable (here, age) and an outcome variable (here, functional connectivity).

Conceptually, the model can be written as:

Functional Connectivity = Intercept + (Slope × Age)

The slope tells us how much functional connectivity is expected to change as we move from one age bin to the next.

Notice that simple linear regression still asks a very similar scientific question to correlation:

Does functional connectivity change with age?

However, regression provides much more information than a correlation coefficient alone.

For each connectivity measure, the regression model estimates:

• B: the unstandardized regression coefficient (slope), indicating how much functional connectivity changes for each increase in age bin.

• SE: the standard error of the slope, describing the uncertainty of the estimate.

• β: the standardized regression coefficient, allowing us to compare the strength of the relationship across different predictors that are measured on different scales. (For example, if you later include Age, Education, and Sex in the same model, β lets you compare which predictor has the strongest association with connectivity, even though they are measured in different units.)

• t and p: statistical tests of whether the estimated slope is significantly different from zero.

• R²: the proportion of variability in functional connectivity that can be explained by age.

In this section, we will fit a separate simple linear regression model for each of our nine functional connectivity measures.

Although these models contain only one predictor (age), they introduce the same statistical framework that we will later extend to multiple linear regression and moderation analysis.

Interpreting the Linear Regression Results¶

Before looking at the plots, let’s first learn how to interpret the regression table.

For each connectivity measure, there are two quantities that are particularly useful:

• β (beta) tells us the direction and strength of the relationship.

• p-value tells us whether there is enough statistical evidence to conclude that the relationship is unlikely to have occurred by chance.

Interpreting β (beta)¶

The sign of β tells us the direction of the relationship.

• A positive β means that connectivity tends to increase as age increases.

• A negative β means that connectivity tends to decrease as age increases.

• A β close to zero suggests little or no relationship.

The magnitude of β tells us how strong the relationship is. Larger values (in absolute value) indicate a stronger association between age and connectivity.

Interpreting the p-value¶

The p-value tells us whether the observed relationship is statistically significant.

In this tutorial, we corrected the p-values using the Bonferroni correction because we tested nine different connectivity measures.

As a general guide:

• Bonferroni-adjusted p < 0.05 suggests evidence for an association.

• Bonferroni-adjusted p ≥ 0.05 suggests that the observed relationship could plausibly have occurred by chance.

With this in mind, we can now examine the regression plots.

The first thing to notice is that most of the regression lines slope slightly upwards (with the exception of DMN–Limbic). This means that, across many of these connectivity measures, functional connectivity tends to increase as we move from younger to older age groups.

However, not all of these increases are equally convincing.

The strongest positive relationships are seen for:

• DMN–FPN connectivity (β = 0.224, Bonferroni-adjusted p = 0.008),

• DMN–VAN connectivity (β = 0.208, Bonferroni-adjusted p = 0.017),

• DMN–DAN connectivity (β = 0.191, Bonferroni-adjusted p = 0.039).

Notice that these three plots have the steepest positive slopes, and they are also the only connectivity measures that remain statistically significant after correcting for multiple comparisons.

This is interesting because these are all higher-order association networks, involved in functions such as attention, executive function, and cognitive control. In this dataset, they appear to become more strongly connected with the Default Mode Network as age increases.

Now compare these with the DMN within DMN plot.

Here, the regression line is almost completely flat (β ≈ 0.02, p = 1.00 after Bonferroni correction). This suggests that connectivity within the Default Mode Network itself does not systematically change across the age groups represented in this dataset.

Similarly, DMN–SMN connectivity show little evidence of an age-related relationship.

Summary¶

So far, these analyses paint a consistent picture: connectivity between the DMN and several higher-order association networks tends to increase across the age groups represented in the LEMON dataset.

However, there is still an important limitation.

So far, every model has considered only one predictor: age.

But participants differ in many other ways besides age. They also differ in sex, education, depression scores, alcohol use, relationship status, and many other behavioral or demographic characteristics.

This naturally leads to our next question:

Does age still predict functional connectivity after accounting for these other participant characteristics?

To answer that question, we now extend simple linear regression to multiple linear regression, where several predictors can be included in the same statistical model.👇

Multiple Linear Regression: Controlling for Additional Variables¶

In the previous section, we used simple linear regression to investigate whether functional connectivity changed with age.

Our model contained one predictor:

• Predictor: Age

• Outcome: DMN–FPN connectivity

Mathematically, a simple linear regression model can be written as:

where:

• β₀ is the intercept (the predicted connectivity when Age = 0),

• β₁ is the regression coefficient (or slope) for age,

• ε represents the remaining unexplained variation.

This equation simply says that we are trying to explain functional connectivity using one predictor: age.

In statistical software such as Python or R, the same model is often written using a shorthand formula notation:

DMN–FPN connectivity ~ Age

The symbol ~ can be read as:

“is modeled as a function of”

or simply

“is predicted by”

So,

DMN–FPN connectivity ~ Age

can be read as:

DMN–FPN connectivity is modeled as a function of age.

Why do we need multiple linear regression?¶

Our simple regression showed that connectivity between the DMN–FPN, DMN–VAN, and DMN–DAN increased with age, with standardized regression coefficients of approximately β ≈ 0.20, and these relationships (but not others) remained statistically significant after correcting for multiple comparisons.

However, age is unlikely to be the only factor influencing brain connectivity.

Participants also differ in many other characteristics, including:

• sex,

• education,

• depression score,

• alcohol consumption,

• relationship status,

• handedness.

These variables may also contribute to differences in functional connectivity.

This naturally raises a new question:

Does age still predict functional connectivity (e.g., DMN-FPN) after accounting for these other participant characteristics?

To answer this, we extend our model by including multiple predictors.

Instead of modeling connectivity using only age,

DMN–FPN connectivity ~ Age

we now fit the model

DMN–FPN connectivity
~
Age
+
Sex
+
Education
+
Hamilton Scale (depression scores)
+
AUDIT (alcohol use disorder test scores)
+
Relationship Status

Mathematically, this becomes

Each predictor has its own regression coefficient (β), allowing the model to estimate the contribution of each variable while accounting for all the others.

For example, the coefficient for Age (β0) now answers the question:

If two participants have the same sex, education, Hamilton score, AUDIT score, relationship status, and handedness, does the older participant still tend to have higher DMN–FPN connectivity?

The participant information used in this model such as relationship status comes from the cleaned metadata table (analysis_meta), which was previously saved as lemon_analysis_metadata.csv.

Coding: Steps¶

Before fitting the regression model, we need to prepare the data.

We will:

• merge the participant metadata with the FC summary measures,

• remove participants with missing values in any of the variables used in the model,

• convert categorical variables (e.g., sex and education) into numeric variables that the regression model can use,

• fit a multiple linear regression predicting DMN–FPN connectivity,

• examine the regression coefficients, confidence intervals, and statistical significance of each predictor.

Interpreting the Multiple Linear Regression¶

Does age remain a significant predictor?¶

Recall that in the previous section, we found that DMN–FPN connectivity increased with age using a simple linear regression.

Here, we extended the model by including several additional variables such as sex, education, depression scores, scores from alcohol use disorders identification test (AUDIT), and relationship status.

This allows us to ask a more specific question:

Does age still predict DMN–FPN connectivity after accounting for these other participant characteristics?

Here, based on the results, age remains a significant predictor!

The regression coefficient for Age remained statistically significant:

• B = 0.0106

• β = 0.243

• p = 0.0045

• 95% CI = [0.0033, 0.0179]

What does this mean?

Imagine comparing two participants who have the same sex, education level, Hamilton score, AUDIT score, and relationship status, but differ in age. The model estimates that the participant in the older age bin is expected to have, on average, higher DMN–FPN connectivity.

This is what is meant by controlling for the other variables. In this analysis, variables such as sex, education, depression score, alcohol use, and relationship status are treated as covariates. Remember, in our research question, we are primarily interested in AGING, which is the main variable of interest. Therefore, variables that are not the main focus of the research question are called “covariates”. By including them in the regression model, we account for their potential influence, allowing us to estimate the relationship between age and DMN–FPN connectivity more accurately.

Notice that the standardized regression coefficient (β = 0.243) is very similar to the value obtained in the simple linear regression (β ≈ 0.224). This suggests that the relationship between age and DMN–FPN connectivity is not explained by differences in sex, education, mood, alcohol use, or relationship status. Instead, age appears to have its own independent association with DMN–FPN connectivity in this dataset.

The other variables were not statistically significant**¶

The remaining predictors all had p-values greater than 0.05, indicating that they were not significantly associated with DMN–FPN connectivity in this model.

For example:

• Sex: p = 0.289

• Education: p = 0.296

• Hamilton Scale: p = 0.159

• AUDIT: p = 0.160

• Relationship Status: p = 0.571

This does not necessarily mean these variables have no relationship with brain connectivity. Rather, in this sample and for this particular model, there was not enough evidence to conclude that they explained additional variation in DMN–FPN connectivity beyond age.

Why do we care about explaining variation?¶

Every participant has a slightly different level of DMN–FPN connectivity. This natural spread in the data is called variation.

The goal of regression is to understand why this variation exists.

For example, some of the variation might be explained by:

• age,

• education,

• mood,

• alcohol use,

• or other biological and environmental factors.

A useful regression model explains as much of this variation as possible. The more variation we can explain, the better we understand which factors are associated with differences in brain connectivity.

The model explained approximately 7.5% of the variation in DMN–FPN connectivity (R² = 0.075). This means that the predictors included in the model collectively accounted for about 7.5% of the differences in DMN–FPN connectivity across participants.

After accounting for the number of predictors in the model, the adjusted R² was 0.046, meaning that approximately 4.6% of the variation remained explained after penalizing the model for its complexity.

Although these values are relatively modest, this is common in neuroscience. Brain connectivity is influenced by many genetic, biological, behavioral, environmental, and measurement-related factors that are not included in a single regression model. Therefore, even models with relatively small R² values can still reveal meaningful relationships. At the same time, one goal of neuroscience research is to better explain more of the variation in our brain. This may involve including or collecting additional variables, such as cognitive performance, genetic factors, lifestyle, or other brain imaging measures, that help us better understand why individuals differ from one another.

Overall, these results suggest that age remains an independent predictor of DMN–FPN connectivity, even after accounting for several demographic and behavioral covariates.

Although we’ve explained about 4.6% of the variation, what other factors might explain the remaining 95.4%?

Moderation Analysis¶

So far, we have found that DMN–FPN connectivity tends to increase with age, even after accounting for several participant characteristics using multiple linear regression.

As neuroscientists, however, we might wonder whether this tells the whole story.

The Default Mode Network (DMN) and the Frontoparietal Network (FPN) have both been implicated in healthy aging, but they have also been linked to depression and other psychiatric conditions. This raises an interesting biological question:

Could the relationship between age and DMN–FPN connectivity depend on a participant’s level of depression?

In other words, perhaps aging affects DMN–FPN connectivity differently for people with low depression scores than for people with higher depression scores.

This type of question is called a moderation analysis.

What is a moderation effect?¶

Imagine two participants who are the same age.

One has very low depression symptoms, while the other has much higher depression symptoms.

Would we expect aging to influence their brain connectivity in exactly the same way?

Perhaps not.

For example:

• among people with low depression scores, DMN–FPN connectivity might increase only slightly with age;

• among people with high depression scores, DMN–FPN connectivity might increase much more rapidly with age.

If this were true, we would say that depression moderates the relationship between age and DMN–FPN connectivity.

Rather than asking

Does age predict connectivity?

we now ask a more specific question:

Does depression change or moderate the relationship between age and DMN–FPN connectivity?

Adding an interaction term¶

Previously, our multiple linear regression model included age and several covariates:

DMN–FPN connectivity
~
Age
+
Sex
+
Education
+
Hamilton Scale
+
AUDIT
+
Relationship Status

To test for moderation, we keep all of these predictors and add one extra term:

DMN–FPN connectivity
~
Age
+
Sex
+
Education
+
Hamilton Scale
+
AUDIT
+
Relationship Status
+
Age × Hamilton Scale

The new term,

Age × Hamilton Scale

is called an interaction term.

It represents the possibility that the effect of age is different at different levels of depression.

Our research question therefore becomes:

After controlling for the same covariates as before, does depression score moderate the relationship between age and DMN–FPN connectivity?

If the interaction term is statistically significant, it suggests that the relationship between age and connectivity depends on depression score.

If the interaction term is not statistically significant, then the relationship between age and connectivity is similar regardless of depression score.

Moderation analysis is widely used in neuroscience because many biological and behavioral relationships are unlikely to be the same for every participant. Instead, one factor may strengthen, weaken, or even reverse the influence of another.

Interpreting the Moderation Analysis¶

As a reminder, our research question was:

Does depression score moderate the relationship between age and DMN–FPN connectivity?

To answer this, we fitted a multiple linear regression model that included:

• Age

• Sex

• Education

• Hamilton Scale (depression score)

• AUDIT

• Relationship Status

• Age × Hamilton Scale (interaction term)

Is the interaction significant?¶

The first thing we should look at is the interaction term, since this directly tests our research question.

The interaction between Age and Hamilton Scale was statistically significant (p < 0.05):

• B = 0.0026

• β = 0.235

• p = 0.026

• 95% CI = [0.0003, 0.0049]

This tells us that the relationship between age and DMN–FPN connectivity changes depending on a participant’s depression score.

Interestingly, neither Age (p = 0.216) nor Hamilton Scale (p = 0.808) was statistically significant on its own.

This may seem surprising at first, but remember that moderation asks a different question.

Instead of asking:

Does age predict connectivity?

or

Does depression predict connectivity?

we are asking:

Does depression change how age is related to connectivity?

A significant interaction suggests that the answer is yes.

One possible interpretation is that the positive relationship between age and DMN–FPN connectivity is stronger for participants with higher depression scores than for those with lower depression scores.

Notice that this does not mean depression itself increases or decreases connectivity. Instead, it suggests that depression changes how strongly age is associated with connectivity.

Do we need multiple-comparison correction?¶

You may notice that the regression output reports a p-value for every predictor in the model.

However, not every coefficient necessarily represents a separate scientific hypothesis.

In this analysis, our primary hypothesis concerns the Age × Hamilton Scale interaction. The remaining variables (sex, education, AUDIT, relationship status, etc.) are included as covariates to account for potential confounding influences.

Because the interaction term is our single pre-specified hypothesis, many neuroscience studies would report its uncorrected p-value.

If, instead, we were testing many different interaction terms or fitting many regression models, then multiple-comparison correction (for example, Bonferroni or False Discovery Rate correction) would become appropriate.

Visualizing the interaction¶

The regression table tells us that the interaction is statistically significant, but it does not show us what that interaction actually looks like.

To better understand the result, we can visualize the data.

Let’s divide participants into three groups based on their Hamilton depression scores:

• Low

• Medium

• High

For each group, we plotted DMN–FPN connectivity against age and fitted a separate regression line.

Notice that this is not another statistical test. Instead, it provides an intuitive way to visualize the interaction estimated by the regression model and helps us answer questions such as:

• Does the relationship between age and DMN–FPN connectivity become stronger as depression scores increase?

• Are the regression slopes noticeably different across the three depression groups?

Interpreting the interaction plot¶

The figure provides a visual illustration of the moderation effect.

Several observations stand out.

First, participants with higher Hamilton scores (green line) appear to show the steepest increase in DMN–FPN connectivity with age.

In contrast, participants with lower Hamilton scores (blue line) show only a relatively small increase in connectivity across the age range.

The medium depression group lies between these two extremes.

These different slopes are exactly what we would expect from a significant interaction term. They suggest that the relationship between age and DMN–FPN connectivity is not constant across all participants, but instead varies according to depression score.

Importantly, this figure should be interpreted as a visualization of the fitted regression model rather than definitive proof on its own. The statistical evidence for the interaction comes from the regression coefficient for the Age × Hamilton Scale term (p = 0.026). The figure simply helps us understand what that significant interaction means in practice.

One possible interpretation is that age-related increases in DMN–FPN connectivity are more pronounced among participants with higher depression scores than among those with lower depression scores.

Quantifying the slopes¶

The interaction plot suggests that the slope relating age to DMN–FPN connectivity differs across depression groups.

Rather than relying on visual inspection alone, we can quantify these slopes directly.

For each Hamilton group (Low, Medium, and High), we will fit a simple linear regression model:

DMN–FPN connectivity ~ Age

This will allow us to estimate:

• the regression slope,

• its confidence interval,

• and the statistical significance of the age effect within each depression group.

Comparing these slopes provides a simple way of understanding how the age–connectivity relationship changes across different levels of depression symptoms.

Interpreting the Slopes¶

To better understand the significant interaction between Age and Hamilton Scale, we estimated the relationship between age and DMN–FPN connectivity separately for participants with Low, Medium, and High depression scores.

The results showed a clear pattern.

For participants with low Hamilton scores, there was a small positive relationship between age and DMN–FPN connectivity (B = 0.0065), but this relationship was not statistically significant (p = 0.132).

Similarly, for participants with medium Hamilton scores, the estimated slope was slightly larger (B = 0.0085), although it also did not reach statistical significance (p = 0.111).

In contrast, participants with high Hamilton scores showed a much steeper positive relationship between age and DMN–FPN connectivity (B = 0.0196), and this association was statistically significant (p = 0.0067).

These findings are consistent with the interaction observed in the multiple regression model. Specifically, they suggest that the positive association between age and DMN–FPN connectivity becomes stronger as depression scores increase.

Altogether, the moderation analysis and the follow-up subgroup analyses allowed us to learn something new about how depression scores may moderate the effect of aging on DMN-FPN interaction!

Tutorial Summary and Next Steps¶

Congratulations! By completing this tutorial, you have learned several of the most commonly used statistical approaches for studying relationships between the brain and behavior.

In particular, you have learned how to:

• quantify associations using correlation,

• model relationships using simple linear regression,

• account for additional variables using multiple linear regression, and

• investigate interaction effects using moderation analysis.

These techniques form the foundation of a large proportion of modern neuroimaging research, and many neuroscience papers rely on these statistical concepts.

Building good research questions¶

One of the most important lessons from this tutorial is that statistics should be driven by scientific questions, not the other way around.

Before building a regression model, ask yourself:

• What am I trying to understand?

• Which variable is my main variable of interest?

• Which other variables might influence this relationship?

• Should these variables be treated as covariates?

• Is there a reason to believe that one variable may moderate another?

A good regression model is motivated by theory, previous literature, or a clearly defined hypothesis.

For example, if your research question is:

Does personality predict functional connectivity?

then personality may be your main predictor, while age and sex may be included as covariates because they are known to influence both personality and brain connectivity.

On the other hand, if your research question is:

How does aging affect functional connectivity?

then age becomes the main predictor, while personality may instead be treated as a covariate.

The same variable can therefore play different roles depending on the research question.

Exploratory versus hypothesis-driven research¶

Not every analysis needs to begin with a strong hypothesis.

Sometimes researchers perform exploratory analyses to search for interesting patterns that may motivate future studies.

However, exploratory analyses require careful interpretation.

If we were to test hundreds or even thousands of possible relationships, some will inevitably appear statistically significant purely by chance.

This is why multiple-comparison correction methods, such as Bonferroni correction or False Discovery Rate (FDR) correction, become increasingly important when performing many statistical tests.

As neuroscientists, our goal is not simply to find statistically significant results. Instead, we aim to identify findings that are:

• biologically plausible,

• statistically robust,

• reproducible,

• and supported by previous evidence.

Developing this critical way of thinking is just as important as learning how to write the analysis code itself.

A suggested exercise¶

Now that you understand the basic statistical workflow, try reading published neuroimaging papers and identify how they built their statistical models.

For example, Simon et al. (2020) investigated whether personality traits were associated with resting-state functional connectivity. They reported findings such as:

“Openness was positively associated with connectivity in the default-mode network, neuroticism was negatively associated with both the ventral and dorsal attention networks, and agreeableness was negatively associated with the dorsal attention network.”

In their statistical models, they first included age and sex as demographic variables before entering the personality measures.

Ask yourself:

• Which variables were the main predictors?

• Which variables were treated as covariates?

• Why might age and sex have been included?

• Can you interpret the regression coefficients reported in their Table 2?

• Can you understand their moderation analysis after completing this tutorial?

These are excellent exercises for developing your ability to critically read the scientific literature.

Try your own research question¶

The LEMON dataset contains many behavioral and demographic measures beyond age.

Try investigating whether resting-state functional connectivity is associated with:

• personality traits,

• cognitive measures,

• depression or anxiety scores,

• lifestyle measures,

• education,

• handedness,

• physiological measurements,

• or any other phenotype available in the dataset.

Can you reproduce findings reported in previous publications?

Can you discover new hypotheses worth investigating?

Continue learning¶

The statistical methods introduced here are only the beginning.

As you continue learning, you may wish to explore topics such as:

• the mathematical derivation of linear regression,

• mediation analysis,

• logistic regression,

• mixed-effects models,

• generalized linear models,

• structural equation modeling,

• Bayesian regression,

• and causal inference.

A deeper understanding of these methods will help you interpret scientific findings more critically and build more sophisticated models for your own research.

Looking ahead¶

In this tutorial, we primarily used linear regression as a tool for association analysis. We asked questions such as:

Is functional connectivity associated with age?

In the next tutorial, we will shift our focus from association to prediction.

Interestingly, the same linear regression model that we used here is also one of the machine learning algorithms. Instead of asking whether two variables are associated, we will ask whether brain connectivity can be used to predict behavioral or demographic measures in unseen participants.