 # Building Multiple Linear Regression Model

Less of a Blog, More of a Note on Model Building

This week we studied about Estimation and Regression. For estimation, we specifically looked at T-Test or Student T-Test, and for regression, we particularly looked at linear regression.

We studied T-Test to evaluate differences in the means. In this process, I also came across a fun fact that T-test was developed by William Gosset. He introduced T-Test in 1908 while he worked at Guinness Brewery in Dublin. It was interesting to find out that he applied his findings using the T-Test to monitor the quality of stout in the production of dark beer. While he wanted to optimize and maintain the standard, he was researching how to make that happen for Guinness and that’s when he came up with T-Test statistics. At first, Guinness did not allow him to publish his research publicly in fear that other competitors might take advantage, and Guinness won’t have competitive advantage against them. But later Guinness agreed to allow Gosset to publish his finding under a pseudonym “Student”. That’s why, T-test is also known as Student T-test.

T-test helps us compare means of two groups. There are two key assumptions for T-test:
1. The samples need to be independent
2. The samples need to be normally distributed (have a bell-curve or symmetry around the mean)
We usually use histogram to check if the distribution of the samples is normally distributed or not, but this week we also studied about a graph- scattered plot known as Q-Q plot, which helps us quickly eyeball the graph, where if the pattern of the scattered plot is an upward straight line, we can say that the samples are normally distributed.

The SAS PROC T-TEST procedure is used to test for the equality of means for a two-sample (independent group) t-test. The typical hypotheses for a two-sample t-test are:
Ho : µ𝟏 = µ𝟐
Ha : µ𝟏 ≠ µ𝟐

Once we run the PROC T-TEST, we get the results – pooled estimator and Satterthwaite value. If the variance is equal, then we look at the p-value of the pooled estimator, otherwise, if the variance is unequal, we look at the Satterthwaite, which is an alternative estimator. If the p-value is <0.05 (5%), we reject the null hypothesis that the means are equal, and we accept the alternative hypothesis that the means are different.

Another thing for T-test is that the class statement must consist of only two levels. It cannot be more than two levels. For example, we can run t-test between wheat beer and lager beer, but we cannot run t-test for 10 different types of beer at once.

Now lets talk about linear regression modeling, it is a foundational tool for prediction, estimation and forecasting across a wide variety of domains and problems. In linear regression, there is a response variable (variable that we want to predict also known as y-variable, or dependent variable), and predictor variable (variable that we will use to explain the response variable, also known as x-variable, independent variable, or explanatory variable).

In statistics, we know that the linear equation has a straight line formula, y=a + bx, where a is the intercept which crosses y-axis, and b is the slope of that line. With this formula, we can determine the linear relationship between two variables. So regression is the statistical technique for finding the best fitted straight line through a set of data, and the resulting straight line is regression line, where the equation is also represented as y= a+bx. But, here, we also get an error, which is the difference between the actual data point (Y) and predicted data point (Y ̂) on the line. Total squared error is simply the sum of those differences.
error=Y-Y ̂
Total squared error: ∑[(Y-Y ̂)]^2

We need to know that the larger the correlation (which we can observe from the scattered plot), the less the error will be. So the accuracy of our linear regression depends on how well the points on the line correspond the actual data points, i.e., the amount of error. When there is a perfect correlation (r = 1 or r = -1), the linear equation fits the data perfectly. So we are looking for the r^2 value or adjusted r^2 value to be closer to 1 so that we can tell that the model is perfectly fitted. For example, if we have adjusted r^2 of 0.61 (61%), we can say 61% of the variance in the dependent variable is explained by the combination of independent variables of our model. We look at adjusted r^2 because it increases only if the new term improves the model more than it would be expected by chance.

Talking about Multiple Linear Regression model, it has to have numeric variables, eg Categories, Nominal, Group etc. variables have to be One-Hot-Encoded which means all of those categories needs to be assigned to numeric variable for its identification to be able to use in the model. We also need to have complete records meaning observations containing nulls are ignored, and since we are dealing with linear regression, there needs to be linear relationship between the outcome variable and the independent variables.

Another thing to note is how to build a multi linear model. There are few steps to take into consideration. At first we explore the data, where we use PROC means, univariate and sgplot to explore numeric and continuous variables, and generate descriptive statistics, and number of counts, number of missing value, mean, median, and discard any columns and observations that have a lot of missing values so that we can prepare or transform our data. Before that, we also run correlation between variables with PROC corr and plot the SGPLOT scatter, box, and histograms to determine the correlation between all the variables. Next step, we plot the distribution and see if it has skewed or normal distribution by looking at the histograms. If we find the variables to have skewed distribution then we use data step to transform those variables to have normal distribution by using LOG(x), LN(x), X^2, 1/X, SQRT(X). Among these whichever method gives us the normal distribution for those variable, we use that method in our model. Simultaneously, we also describe rules to identify and exclude outliers and extremes that affects our mean.

We also use PROC freq and sgplot to check the frequency to explore categorical / nominal variables. Then we look at preparing or transforming our continuous variable, basically replacing missing values with mean or median, and applying normalization model. We use that to prevent our results from being skewed because if we input skewed data, our model won’t predict good results. As we know Garbage In Garbage Out, we want to make sure to input good data to build and train our model, so it produces better results.

When building our model we need to remember encoding categorical/nominal variables to 0/1s. We use PROC sql CASE WHEN or DATA step IF/THEN to do the encoding. Once we have transformed and prepared our dataset, we will partition our whole dataset into training set (75%) or test set (25%). We will iteratively train our model using training set. For evaluating the model, we check for the following steps after running PROC Reg:

So far, I have managed to understand PROC REG procedure to build multiple linear regression model. However, there are other alternatives to PROC REG procedure in building a regression model. These alternative procedures, I have yet to explore further. For reference, below are the snapshot of comparisons between the procedures provided in the class.

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