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# ST313 Applied Regression Models Assignment Sample NUI Galway Ireland

ST313 Applied Regression Models is a course offered by National University in Ireland, Galway. This course covers topics such as simple and multiple regression, coefficient interpretation, model selection and evaluation, diagnostics, and residual analysis. The course also covers more advanced topics such as transformations, interactions, splines, and Polynomial Regression models on time series data.

This course is designed for students who have a strong interest in learning about regression methods and applying them to real data sets. The material covered in this course will be beneficial for students interested in pursuing careers in data analysis or working with data regularly.

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In this section, we are describing some assigned activities. These are:

### Assignment Activity 1: Calculate and interpret correlations between variables and make inferences about relationships.

There are a few different ways to calculate correlations between variables, but the most common is Pearson’s correlation coefficient. This measures the linear relationship between two variables and can range from -1 (perfect negative correlation) to 1 (perfect positive correlation).

Interpreting the results of a correlation can be tricky because it’s important to remember that correlation only measure relationships – they don’t tell us anything about causation. In other words, just because two variables are correlated doesn’t necessarily mean that one causes the other. There could be another third variable that causes both of them, or the relationship could be reversed (i.e. instead of X causing Y, Y is causing X), or there could be no causal relationship at all.

With that said, some general rules of thumb can be helpful when interpreting the results of a correlation:

• A correlation coefficient of 0 means that there is no linear relationship between the two variables.
• A correlation coefficient close to 1 (or -1) indicates a strong linear relationship.
• A positive correlation coefficient (negative) indicates a positive (negative) relationship – i.e. as one variable increases, the other variable also increases (decreases).
• The strength of the linear relationship is determined by how close the correlation coefficient is to 1 (or -1). The closer it is, the stronger the relationship.
• The sign of the correlation coefficient tells us the direction of the relationship.

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### Assignment Activity 2: Formulate a linear regression model, calculate estimated coefficients and make statistical inferences on the fitted model using both parameter estimates and the ANOVA table.

To formulate a linear regression model, we first need to identify our dependent and independent variables. In this case, our dependent variable is “age” and our independent variable is “calories consumed”. We then use the following equation to calculate our estimated coefficients:

where “b 0 ” is the intercept (or the constant), “b 1 ” is the slope of the line, and “e” is the error term. We can then use this equation to make statistical inferences on the fitted model using both parameter estimates and test statistics. For example, we can test whether or not the slope is statistically significant by running a t-test. If the t-test statistic is greater than the critical value, then we can conclude that the slope is statistically significant.

### Assignment Activity 3: Obtain fitted values and predictions at new data points, together with associated confidence intervals.

When fitting a model to data, it is often of interest to obtain fitted values and predictions at new data points, together with associated confidence intervals. This can be done using the predict() function in R.

For example, suppose we have fit a linear regression model to some data. We can use the predict() function to obtain fitted values and predictions for new data points as follows:

fitted_values <- predict(model, new_data)

predictions <- predict(model, new_data, interval = “predict”)

This will give us the fitted values and predictions at the new data points in the ‘new_data’ object. The ‘interval’ argument tells R to calculate the prediction intervals along with the predictions.

Once we have the predictions, we can then use them to obtain confidence intervals for the predictions using the content() function:

conf_int <- confint(model, new_data)

This will give us the confidence intervals for the predictions at the new data points in the ‘new_data’ object.

### Assignment Activity 4: Calculate regression diagnostics and use these to check model assumptions, including linearity, normality, constant variance, independence, and the presence of outliers and influential points.

There are several ways to calculate regression diagnostics and check model assumptions. Perhaps the simplest way is to simply plot the data and look for patterns. This can give you a good visual indication of whether or not the assumption of linearity is met, for example. Other diagnostic measures may be more mathematically complex, but they all ultimately serve the same purpose: to help you check that your data meets the necessary assumptions for reliable regression analysis.

Some of the most important assumptions to check for are linearity, normality, constant variance, independence, and the presence of multicollinearity. Let’s take a brief look at each of these in turn.

Linearity refers to the relationship between independent variables and the dependent variable. For a linear regression model to be reliable, this relationship should be linear. That is, the dependent variable should change in a constant, or nearly constant, manner as the independent variable is varied. If this is not the case, then the model may be inaccurate.

Normality refers to the distribution of the residuals. For a linear regression model to be reliable, the residuals should be normally distributed. That is, they should follow a bell-shaped curve. If this is not the case, then the model may be inaccurate.

Constant variance refers to the variability of the dependent variable. For a linear regression model to be reliable, the dependent variable should have a constant variance. That is, it should not increase or decrease in variability as the independent variables are varied. If this is not the case, then the model may be inaccurate.

Independence refers to the relationship between the residuals. For a linear regression model to be reliable, the residuals should be independent. That is, they should not be correlated with one another. If this is not the case, then the model may be inaccurate.

The presence of outliers and influential points can also adversely affect the accuracy of a linear regression model. Outliers are points that lie far from the rest of the data, while influential points are points that have a large influence on the regression line. If either of these is present, then the model may be inaccurate.

### Assignment Activity 5: Formulate a multiple regression model and specify this in matrix form.

A multiple regression model can be written in matrix form as:

Y = Xβ + ε,

where Y is the dependent variable, X is the matrix of explanatory variables, β is the vector of regression coefficients, and ε is the error term.

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### Assignment Activity 6: Use Principal Components Analysis to reduce the dimensionality of a complex data set.

Principal Component Analysis is a technique for reducing the dimensionality of data. This can be helpful when you have a lot of data and you want to make sense of it, or when you want to simplify the data so that it’s easier to work with.

When you reduce the dimensionality of data, you do so by identifying the important features (or dimensions) in the data. You can then discard the less important features, or keep them if they’re still useful. This process makes it easier to understand and work with the data, and can also improve performance when performing tasks such as machine learning or predictive modeling.

To reduce the dimensionality of a data set using Principal Component Analysis, you first need to identify the variance in the data. This can be done by computing the covariance matrix. The covariance matrix is a square matrix that contains the variances of the variables in the data set.

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