Lab Assignments

Part 1: Matching Law

This week, we are focusing on the generalized matching equation (GME). The purpose of this lab is to use your programming skills to fit the GME to participant data and interpret the outputs. In the folder, there are 10 hypothetical datasets with participant data common to matching law experiments. Your job is what a researcher's job might be. That is, see how well the data are fit by the GME for each individual as well as any trends that might be worth discussing at the group level.

The following packages will likely help accomplish this task: pandas, numpy, matplotlib, seaborn, and scipy (linregress function). These are not the only ways you might accomplish the goal, but they certainly have everything you need.

Part 2: Discounting

We are also focusing on the family of equations within the discounting area of the literature. The purpose of this part of the lab is to use your programming skills to fit the hyperbolic, hyperboloid, and Area Under the Curve equations to participant data and interpret the outputs.

In the folder, there is a single dataset with participant data common to discounting experiments. Specifically, each row has a set of indifference points relative to seven different indifference points (columns) along with information about the amount and commodity specific to those indifference points.

Your job is what a researcher's job might be. That is: (1) see how well the data are fit by the hyperbolic and hyperboloid models; (2) once fit, see if there are any trends in discounting relative to changes in amount, commodity, or sign (gain vs. loss).

The following packages will likely help accomplish this task: pandas, numpy, matplotlib, seaborn, scipy (for fitting and calculating AUC), and scikit-learn (for r2 values).

As a bonus, in the folder, you will also see a params_data.csv. These were the raw k and s parameters used to create each participant's data while also adding noise pulled from the normal distribution between -10 and 10. You could compare how well your package derived the original parameters when noise is involved.

Demand Lab

This week, we are focusing on the family of equations within the demand area of behavior analysis. The purpose of this lab is to use your programming skills to explore demand. In this assignment, you will fit the exponential demand equation proposed by Hursh & Silberberg (2008) to participant consumption data.

In this folder, there is a single dataset with participant data common in demand experiments. Specifically, each row has a single participant with consumption at a range of price points (i.e., $0.01,$0.05, $0.25,$1, $5,$10, $33,$100, $250,$500, $1000). The Q0 and alpha values used to generate each participant's data are also present. However, noise was added once generated. Your job is to calculate Q0 from the participant's data directly, calculate the k value from the dataset as a whole, and then systematically loop through each participant and fit the exponential equation, estimating alpha for each participant.

General Steps

1. Read in the dataset
2. Transform the dataset to be in long form
3. Transform the price to a numeric value
4. Calculate the parameter k or choose its constant value
5. Create a function that calculates the consumption Q using the exponential demand equation
6. Apply the function to each row in the dataset to identify the predicted consumption
7. Graph and analyze the results.
8. Calculate the goodness of fit metrics and display them to interpret the model's goodness of fit

Things to Watch Out For

• The log of 0 is undefined because there's no number you can raise that will give you zero.
• Often to avoid the log(0) issue, people will add a constant to all values in the dataset before taking the log (i.e., all numbers in the series are adjusted up by 0.001 before the log transform is applied).

Reference

Hursh, S. R., & Silberberg, A. (2008). Economic demand and essential value. Psychological Review, 115(1), 186-198. https://doi.org/10.1037/0033-295X.115.1.186

Associative Learning Models Lab

This week, we focus on respondent conditioning -- stimulus-stimulus learning -- and the quantitative models that describe how organisms learn predictive relationships between events. Both models in this lab are recursive: the state of the system at trial t (the associative strengths, and for Mackintosh the attention weights) becomes part of the input at trial t+1. Formally:

X(t+1) = f(X(t), I(t))

where X(t) is the state at time t, I(t) is the input or event at time t (a CS, the US, a prediction error), and f is the transition rule. Because both parts are simulations, no data files are required.

Part 1: The Rescorla-Wagner Model

The Rescorla-Wagner (1972) model is the foundational quantitative account of associative learning. Its central claim is that learning is driven by prediction error: associative strength changes only to the extent that the outcome of a trial differs from what the organism already expected. For each stimulus present on a trial,

ΔV_i = α_i * β * (λ - V_total), where V_total = Σ V_i

Here α_i is the salience of stimulus i, β is the learning rate of the US, λ is the asymptote supported by the US (1.0 when it occurs, 0.0 when it does not), and V_total is the summed associative strength of all stimuli present on that trial.

Your task is to implement this update rule as a function and use it to reproduce four classic phenomena that all emerge from the shared prediction-error term:

• Blocking: a CS that already predicts the US prevents learning to a newly added CS.
• Overshadowing: when two CSs are trained together, the more salient one captures more associative strength.
• Overexpectation: two separately conditioned CSs, then trained together, both lose strength because their summed prediction overshoots λ.
• Conditioned inhibition: a CS paired with the absence of an otherwise-predicted US acquires negative associative strength.

Hints:

• Represent each trial as the set of stimuli present plus the US asymptote λ for that trial. Build the phenomenon by choosing the right sequence of trials across phases.
• Compute V_total from the stimuli present before updating, then apply ΔV to each present stimulus.

Part 2: The Mackintosh Model (Recursive Associability)

The Mackintosh (1975) model extends prediction-error learning with a dynamic associability (attention) term. A stimulus that predicts the US better than its competitors gains associability across trials, while a poorer predictor loses it. This lets the model capture attentional phenomena that fixed-salience Rescorla-Wagner cannot.

Your task is to implement the Mackintosh update rules for association strength V and associability alpha, then write simulation functions for (a) basic conditioning, (b) overshadowing, and (c) blocking, and visualize how V and alpha evolve across trials for a range of learning rates (theta).

Hints:

• Recursive nature: both the associability (alpha) and the association strength (V) update based on their previous values.
• Phenomenon validation: your outputs should show basic conditioning (V rises toward λ while alpha adjusts), overshadowing (CS1 with higher alpha gains more V than CS2), and blocking (CS2 shows minimal learning when CS1 already predicts the US).

References

• Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement.
• Mackintosh, N. J. (1975). A theory of attention: Variations in the associability of stimuli with reinforcement. Psychological Review, 82(4), 276-298.

Behavioral Momentum and Response Persistence Lab

Behavioral momentum theory (Nevin, 1992; Nevin & Shahan, 2011) provides a quantitative framework for understanding why behavior persists in the face of disruption. The central finding is that behavior maintained by higher rates of reinforcement is more resistant to change, analogous to how a more massive object in motion is harder to stop. A key strength of the theory is that this same idea is expressed across different disruptors, each with its own equation.

Part 1: Prefeeding

You will work with data from a simulated multiple-schedule experiment in which subjects responded under both rich and lean reinforcement components. After stable baselines were established, responding was disrupted via prefeeding at graded amounts (0, 25, 50, 75, and 100g). Your task is to calculate proportion of baseline responding under each disruption level, fit the behavioral momentum equation to describe how responding decreases as a function of disruption intensity, and compare resistance to change across the two reinforcement contexts.

The key equation you will fit is:

log(Bx/B0) = -x * c / (r * S)

where Bx is the disrupted response rate, B0 is the baseline rate, x is the disruption level, c is sensitivity to disruption, r is the reinforcement rate in the component, and S captures the stimulus-reinforcer relation.

Part 2: Other Disruptors

Prefeeding is only one way to disrupt responding. In this part you will fit two further forms from Nevin and colleagues to data where the disruptor is extinction and where it is alternative reinforcement. The required reading (Nevin et al., 1983) develops these forms; you will dig into the equations and fit them per condition.

• Resistance to extinction: Bt/B0 = 10^(-t * (c + d*r) / sqrt(r)), fit for c, d, r, and B0.
• Alternative reinforcement: Bx/B0 = 10^(-p * Ra / sqrt(r + Ra)), fit for p, r, and B0.

Hints for Part 2:

• Parameter bounds matter. For extinction: c and d should be positive, r should be > 0.1, and B0 should be a reasonable baseline rate. For alternative reinforcement: p should be positive, r should be > 0.1.
• Initial guesses. B0 should be close to the baseline response rate in the data; r is the reinforcement rate (look at the experimental conditions); c, d, and p are scaling parameters (start with small positive values).

Through these exercises you will gain hands-on experience with the quantitative tools behavioral momentum theory offers for predicting and interpreting response persistence across qualitatively different disruptors.

Model Comparisons Lab

This week, we focus broadly on the topic of model comparisons. This week is less hands-on and more about ensuring that you understand the tools at your disposal. Models generally come in two flavors: regression (continuous or ordinal output) and classification (categorical output). The readings this week highlighted the many different ways we can think about comparing how close our model outputs are to the observed data (i.e., the various fit metrics).

When comparing models, we are choosing one (or more) fit metrics and seeing which model(s) lead to better fit metrics. The loss metric(s) we choose are often determined by our data, the audience, historical precedence, and potentially the downstream deployment environment. Unfortunately, there's no easy decision tree for making this decision. The "right" choice is often determined by thinking critically about your situation, what the model is doing, and how you can interpret the loss metric relative to the "meaning" of the data you are modeling. The nice thing is that choosing a loss metric is the hard part. Once a metric is chosen, comparing models is easy.

The only wrinkle is that models with a greater number of parameters have greater flexibility and, therefore, tend to have lower loss metric values. But the sacrifice is complexity and potentially transparency and explainability. This is where the information criteria metrics are useful (e.g., BIC, AIC, AICc). The idea with these is to penalize models for having more parameters to try to "even the scales." You should always include some measure of information criteria if you are comparing models with different numbers of parameters.

Part 1: Discounting Model Comparison (Regression)

The first dataset contains discounting data from Module 3. This dataset will allow you to compare the following models of discounting: exponential, hyperbolic, Myerson and Green's hyperboloid, and Rachlin's hyperboloid. For every row and for each of the four models, you should obtain the following fit metrics: r-squared, MAE, RMSE, AIC, BIC, and AICc.

Part 2: Challenging Behavior Prediction (Classification)

The second dataset is a hypothetical dataset for a classification task. It was randomly created to allow you to predict whether or not challenging behavior will occur during a session (yes=1, no=0). To introduce yourself to different modeling techniques, try fitting the following: logistic regression, decision tree, random forest, and a support vector machine. You should obtain the following fit metrics: accuracy, precision, recall, F1 score, Matthews Correlation Coefficient (MCC), and ROC-AUC.

Useful Packages

scipy.optimize, scipy.stats, sklearn.metrics, sklearn.linear_model, sklearn.tree, sklearn.ensemble, and sklearn.svm.

How to Construct a Model Lab

This week, we focus on how to construct a model for some phenomenon you are interested in describing quantitatively. The chapters focused on how to build lifecycle diagrams and flow diagrams to start mapping out what variables are in your model and how they might relate. As with past chapters, we can then use the model we created to describe data we have collected, and use various techniques to improve those model fits (e.g., changing mathematically how the different variables relate).

From there, the chapters also covered how we might create a recursive model, a difference model, and a differential model, which allows us to run simulations to get a better feel for our models. This can be incredibly useful to determine when our model might make illogical predictions (e.g., response rates less than zero; exponentially growing response rates). It can also allow us to visualize how different parameters of our model influence our behavioral predictions.

During the in-class demo, we walked through two examples of how to do this for lever pressing under a random ratio (RR) schedule and salivary responding to a conditioned stimulus (CS). We also walked through how we can convert that equation into code to either fit data that we have collected or to run simulations. In each case, the models were one-variable models.

Assignment

Identify two different environment-behavior relations that you are interested in modeling. For each, create:

1. A life-cycle diagram
2. The basic equation relating environmental variables to behavior
3. The set of recursive, difference, and differential equations
4. The code to fit the model to a fake dataset
5. Simulations varying the parameters for each of the equations in step 3

Probability Theory and Probabilistic Models Lab

Probabilistic reasoning is fundamental to drawing inferences from behavioral data. In applied behavior analysis, practitioners routinely make judgments about the function of problem behavior based on patterns observed in functional analysis (FA) conditions. Typically these judgments rely on visual analysis, but Bayesian methods offer a complementary, quantitative approach to updating beliefs about behavioral function as data accumulate.

In this lab, you will implement a Bayesian updating procedure for functional assessment. Starting from a uniform prior over four possible functions (attention, escape, tangible, automatic), you will sequentially incorporate observed FA session outcomes to derive posterior probability distributions over functions. You will also implement a Monte Carlo simulation to estimate confidence intervals for a behavioral parameter, giving you practical experience with simulation-based inference. Together, these exercises illustrate how probability theory can formalize the reasoning that behavior analysts already engage in when interpreting assessment data.

Multilevel Modeling and Time-Series Forecasting Lab

Behavioral data are almost always nested: responses are nested within sessions, sessions are nested within participants, and participants may be nested within groups or settings. Ignoring this nesting structure when fitting statistical models can lead to biased parameter estimates and artificially narrow confidence intervals.

In this lab you will work with a dataset containing session-level response rates from eight participants, each observed across twenty sessions under varying reinforcement rates. You will begin by fitting a single-level ordinary least squares regression that ignores the nesting of sessions within participants. You will then fit a multilevel (mixed-effects) model that accounts for participant-level variation in both intercepts and slopes, compare the two approaches using information criteria, and interpret the fixed and random effects in behavioral terms.

As an optional extension, you will decompose the session-by-session data for a single participant into trend and residual components, providing a brief introduction to time-series thinking that will be developed further in later weeks.

Assignment

1. Load and explore the nested behavioral dataset.
2. Fit an OLS regression pooling all data and examine the residuals.
3. Fit a random-intercept mixed-effects model using statsmodels.formula.api.mixedlm.
4. Extend to a random-intercept-and-slope model.
5. Compare single-level and multilevel models using AIC and BIC.
6. Interpret the fixed effects and random effects in the context of the matching law and reinforcement rate-response rate relations.
7. Visualize participant-level regression lines.
8. (Optional) Perform a simple time-series decomposition on one participant's data.

Dynamical Systems Models Lab

Many behavioral processes unfold over time in ways that are well described by differential equations. The logistic ordinary differential equation (ODE), dx/dt = r * x * (1 - x/K), provides a simple but powerful model of acquisition: responding starts slowly, accelerates as the behavior contacts reinforcement, and then decelerates as it approaches a carrying-capacity asymptote.

In this lab you will implement the logistic ODE numerically using Euler's method, explore how the parameters r (growth rate) and K (carrying capacity) shape the acquisition curve, find the equilibrium points analytically, perform a linear stability analysis, and construct a phase portrait. You will then fit the analytical solution of the logistic equation to empirical acquisition data using nonlinear least squares, comparing the fitted curve to the raw data.

Assignment

1. Implement Euler's method for the logistic ODE and plot the resulting acquisition curves.
2. Vary the initial condition x0 and the parameters r and K to see how they affect the shape and speed of acquisition.
3. Find the equilibrium points of the logistic ODE analytically.
4. Perform a linear stability analysis by evaluating f'(x*) at each equilibrium.
5. Create a phase portrait (dx/dt vs. x) with directional arrows.
6. Load the empirical acquisition dataset and fit the logistic analytical solution using scipy.optimize.curve_fit.
7. Plot the fitted curve alongside the data and report the estimated parameters.
8. Discuss the biological and behavioral interpretation of r and K.

Computational Models Lab

This week bridges the descriptive models of behavior from earlier in the course with process-level computational models that specify how behavior is generated. Rather than fitting equations to existing data, you will build an agent that learns from scratch through trial-and-error interaction with a simulated concurrent VI-VI schedule.

The core question is whether a simple reinforcement learning algorithm -- Q-learning -- produces steady-state choice allocation that resembles the matching law. If it does, this suggests that matching may be an emergent property of basic reinforcement learning dynamics rather than a separate behavioral principle that organisms "follow."

In the lab notebook, you will implement both the environment (concurrent VI 30-s vs VI 60-s schedule) and the agent (Q-learning with softmax action selection). You will then run simulations across multiple sessions, sweep key parameters (learning rate, discount factor), and compare the agent's behavior to the generalized matching equation from Week 2. This exercise illustrates how computational models can serve as mechanistic accounts that give rise to the molar regularities captured by descriptive models.

Assignment

Complete all tasks in the Jupyter notebook. Each task builds on the previous one. Pay particular attention to the final discussion, where you reflect on the relationship between descriptive models (matching law) and process models (Q-learning).

Machine Learning and Artificial Intelligence Lab

This week introduces supervised machine learning as a tool for prediction and classification in behavior science. The specific application is classifying the function of problem behavior from functional analysis (FA) summary data -- a task that clinicians perform routinely based on visual inspection of FA graphs.

You will work with a simulated dataset of 60 participants, each characterized by response rates across four FA conditions (attention, escape, tangible, and play/control). Your goal is to build a classifier that can predict the behavioral function from these features. You will start with a single decision tree, which is highly interpretable and mirrors the kind of rule-based reasoning clinicians use (e.g., "if the rate in the attention condition is elevated relative to play, the function is likely social-positive reinforcement").

From there, you will explore overfitting by comparing training and test accuracy across different tree depths, then build a random forest to see whether an ensemble of trees improves predictive performance. The lab concludes with a discussion of the prediction-explanation tradeoff: more complex models may predict better, but simpler models are easier to interpret and communicate to practitioners.

Assignment

Complete all tasks in the Jupyter notebook. You will need scikit-learn, pandas, and matplotlib. Focus not only on model accuracy but on understanding why the models make the predictions they do.