I conducted a regression analysis on a dataset with 354 data points, aiming to predict a target variable ‘z’ using two predictor variables ‘x1’ and ‘y’ through a linear model:

**Linear Model**: z = b0 + b1*x1 + b2*y + e

My initial plan was to split the data into a training set and a test set to evaluate the model’s performance. However, given the limited amount of data, it didn’t seem practical to do so. Instead, I decided to employ k-fold cross-validation, specifically a 5-fold cross-validation, to assess the accuracy of the linear model.

**Quadratic Equation**: Additionally, I also explored the possibility of fitting a quadratic equation to the data to capture potential non-linear relationships:

**Quadratic Model**: z = b0 + b1*x1 + b2*y + b3*x1^2 + b4*x1*y + b5*y^2 + e

I planned to use cross-validation to compare the performance of the linear and quadratic models, which would help me determine whether a more complex model is warranted given the dataset.

**Mean Square Error (MSE) vs. Model Complexity**: To evaluate model complexity, I intended to compute the Mean Square Error (MSE) for both the linear and quadratic models across different levels of complexity. This would involve incrementally adding higher-order terms (e.g., quadratic terms) to the models and observing how MSE changes as complexity increases. The goal is to identify the model complexity that results in the lowest MSE, which signifies the best trade-off between bias and variance.

**Example and Test Data with 5-Fold Cross-Validation**: For the cross-validation process, I would randomly split the dataset into five equally-sized subsets. Then, I’d train and test the models five times, using each subset as the test set once while the remaining four subsets serve as the training data for each iteration. This process allows me to obtain five different MSE values for each model, which I can then average to get a more robust estimate of model performance.

If you have specific data or need further assistance with the implementation of this approach, please provide the dataset, and I can help you with the actual calculations and code if needed.