# Covariance and Correlation

Covariance measures the degree to which two variables co-vary (i.e. vary/ changes together). If the greater values of one variable (say, $X_i$) correspond with the greater values of the other variable (say, $X_j$), i.e. if the variables tend to show similar behaviour, then the covariance between two variables ($X_i$, $X_j$) will be positive. Similarly if the smaller values of one variable correspond with the smaller values of the other variable, then the covariance between two variables will be positive. In contrast, if the greater values of one variable (say, $X_i$) mainly correspond to the smaller values of the other variables (say, $X_j$), i.e. both of the variables tend to show opposite behaviour, then the covariance will be negative.

In other words, for positive covariance between two variables means they (both of the variables) vary/changes together in the same direction relative to their expected values (averages). It means that if one variable moves above its average value, then the other variable tend to be above its average value also. Similarly, if covariance is negative between the two variables, then one variable tends to be above its expected value, while the other variable tends to be below its expected value. If covariance is zero then it means that there is no linear dependency between the two variables. Mathematically covariance between two random variables $X_i$ and $X_j$ can be represented as
$COV(X_i, X_j)=E[(X_i-\mu_i)(X_j-\mu_j)]$
where
$\mu_i=E(X_i)$ is the average of the first variable
$\mu_j=E(X_j)$ is the average of the second variable

\begin{aligned}
COV(X_i, X_j)&=E[(X_i-\mu_i)(X_j-\mu_j)]\\
&=E[X_i X_j – X_i E(X_j)-X_j E(X_i)+E(X_i)E(X_j)]\\
&=E(X_i X_j)-E(X_i)E(X_j) – E(X_j)E(X_i)+E(X_i)E(X_j)\\
&=E(X_i X_j)-E(X_i)E(X_j)
\end{aligned}

Note that, the covariance of a random variable with itself is the variance of the random variable, i.e. $COV(X_i, X_i)=VAR(X)$. If $X_i$ and $X_j$ are independent, then $E(X_i X_j)=E(X_i)E(X_j)$ and $COV(X_i, X_j)=E(X_i X_j)-E(X_i) E(X_j)=0$.

## Covariance and Correlation

Correlation and covariance are related measures but not equivalent statistical measures. The correlation between two variables (Let, $X_i$ and $X_j$) is their normalized covariance, defined as
\begin{aligned}
\rho_{i,j}&=\frac{E[(X_i-\mu_i)(X_j-\mu_j)]}{\sigma_i \sigma_j}\\
&=\frac{n \sum XY – \sum X \sum Y}{\sqrt{(n \sum X^2 -(\sum X)^2)(n \sum Y^2 – (\sum Y)^2)}}
\end{aligned}
where $\sigma_i$ is the standard deviation of $X_i$ and $\sigma_j$ is the standard deviation of $X_j$.

Note that correlation is the dimensionless, i.e. a number which is free of measurement unit and its values lies between -1 and +1 inclusive. In contrast covariance has a unit of measure–the product of the units of two variables.

# Binary Logistic Regression Minitab Tutorial

Binary Logistic Regression is used to perform logistic regression on a binary response (dependent) variable (a variable only that has two possible values, such as presence or absence of a particular disease, this kind of variable is known as dichotomous variable i.e binary in nature).

Binary Logistic Regression can classify observations into one of two categories. These classifications can give fewer classification errors than discriminant analysis for some cases.

The default model contains the variables that you enter in Continuous predictors and Categorical predictors. You can also add interaction and/or polynomial terms by using the tools available in model sub-dialog box.

Minitab stores the last model that you fit for each response variable. This stored models can be used to quickly generate predictions, contour plots, surface plots, overlaid contour plots, factorial plots, and optimized responses.

To perform a Binary Logistic Regression Analysis in Minitab, follow the steps given below. It is assumed that you have already launched the Minitab software.

Step1:  Choose Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model.

Fit Binary Logistic Regression

Step2:  Do one of the following:

If your data is in raw or frequency form, follow these steps:

Response in Binary Logistic Regression (Frequency Format)

1. Choose Response in binary response/frequency format, from combobox on top
2. In Response text box, enter the column that contains the response variable.
3. In Frequency text box, enter the optional column that contains the count or frequency variable.

If you have summarized data, then follow these steps:

Response in Binary Logistic Regression (Trial Format)

1. Choose Response in event/trial format, from combobox on top of the dialog box.
2. In Number of events, enter the column that contains the number of times the event occurred in your sample at each combination of the predictor values.
3. In Number of trials, enter the column that contains the corresponding number of trials.

Step4:  In Continuous predictors, enter the columns that contain continuous predictors. In Categorical predictors, enter the columns that contain categorical predictors. You can add interactions and other higher order terms to the model.

Step5:  If you like, use one or more of the dialog box options, then click OK.

The following are options available in the main dialog box of Minitab Binary Logistic Regression:

Response in binary response/frequency format: Choose if the response data has been entered as a column that contains 2 distinct values i.e as a dichotomous variable.
Response: Enter the column that contains the response values.
Response event: Choose which event of interest the results of the analysis will describe.
Frequency (optional): If the data are in two columns i.e. one column that contains the response values and the other column that contains their frequencies then enter the column that contains the frequencies.
Response in event/trial format: Choose if the response data are two columns – one column that contains the number of successes or events of interest and one column that contains the number of trials.
Event name: Enter a name for the event in the data.
Number of events: Enter the column that contains the number of events.
Number of trials: Enter the column that contains the number of nonevents.
Continuous predictors: Select the continuous variables that explain changes in the response. The predictor is also called the X variable.
Categorical predictors: Select the categorical classifications or group assignments, such as type of raw material, that explain changes in the response. The predictor is also called the X variable.

Step 6: To stores diagnostic measures and characteristics of the estimated equation click Storage… button.

Binary Logistic Regression Storage Dialog Box

# Multivariable / Multiple Regression

Multiple regression (a regression having multi-variable) is referred as a regression model having more than one predictor (independent and explanatory variable) to explain a response (dependent) variable. We know that in simple regression models has one predictor used to explain a single response while for case of multiple (multivariable) regression models, more than one predictor in the models. Simple regression models and multiple (multivariable) regression models can further be categorized as linear or non-linear regression models.

Note that linearity does not based on predictors or addition of more predictors in simple regression model, it is referred to the parameter of variability (parameters attached with predictors). If the parameters of variability having constant rate of change then the models are referred to as linear models either it is a simple regression model or multiple (multivariable) regression models. It is assumed that the relationship between variables is considered as linear, though this assumption can never be confirmed for case of multiple linear regression. However, as a rule, it is better to look at bivariate scatter diagram of the variable of interests, you check that there should be no the curvature in the relationship.

Multiple regression also allows to determine the overall fit (which is known as variance explained) of the model and the relative contribution of each of the predictors to the total variance explained (overall fit of the model). For example, one may be interested to know how much of the variation in exam performance can be explained by the following predictors such as revision time, test anxiety, lecture attendance and gender “as a whole”, but also the “relative contribution” of each independent variable in explaining the variance.

A multiple regression model have the form

$y=\alpha+\beta_1 x_1+\beta_2 x_2+\cdots+\beta_k x_k+\varepsilon$

Here y is continuous variables, x’s are known as predictors which may be continuous, categorical or discrete. The above model is referred to as a linear multiple (multivariable) regression model.

For example prediction of college GPA by using, high school GPA, test scores, time gives to study and rating of high school as predictors.

# Logistic regression Introduction

Logistic regression was introduced in 1930s by Ronald Fisher and Frank Yates and was first proposed in 1970s as an alternative technique to overcome limitations of ordinary least square regression in handling dichotomous outcomes. It is a type of probabilistic statistical classification model which is non-linear regression model, can be converted into linear model by using a simple transformation. It is used to predict a binary response categorical dependent variable, based on one or more predictor variables. That is, it is used in estimating empirical values of the parameters in a model. Here response variable assumes value as zero or one i.e. dichotomous variable. It is the regression model of b, a logistic regression model is written as

$\pi=\frac{1}{1+e^{-[\alpha +\sum_{i=1}^k \beta_i X_{ij}]}}$

where $\alpha$ and $\beta_i$ are the intercept and slope respectively.

So in simple words, logistic regression is used to find the probability of the occurrence of the outcome of interest.  For example if we want to find the significance of the different predictors (gender, sleeping hours, took part in extracurricular activities, etc.), on a binary response (pass or fail in exams coded as 0 and 1), for this kind of problems we used logistic regression.

By using a transformation this nonlinear regression model can be easily converted into linear model. As $\pi$ is the probability of the events in which we are interested so if we takes the ratio of the probability of success and failure then the model become linear model.

$ln(y)=ln(\frac{\pi}{1-\pi})$

The natural log of odds can convert the logistics regression model into linear form.

References:

# Introduction Odds Ratio

Medical students, students from clinical and psychological sciences, professionals allied to medicine enhancing their understanding and learning of medical literature and researchers from different fields of life usually encounter Odds Ratio (OR) throughout their careers.

Odds ratio is a relative measure of effect, allowing the comparison of the intervention group of a study relative to the comparison or placebo group. When computing Odds Ratio, one would do:

• The numerator is the odds in the intervention arm
• The denominator is the odds in the control or placebo arm= OR

If the outcome is the same in both groups, the ratio will be 1, implying that there is no difference between the two arms of the study. However, if the OR>1, the control group is better than the intervention group while, if the OR<1, the intervention group is better than the control group.

The ratio of the probability of success and failure is known as odds. If the probability of an event is $P_1$ then the odds is:
$OR=\frac{p_1}{1-p_1}$

The Odds Ratio is the ratio of two odds can be used to quantify how much a factor is associated to the response factor in a given model. If the probabilities of occurrences an event are $P_1$ (for first group) and $P_2$ (for second group), then the OR is:
$OR=\frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$

If predictors are binary then the OR for ith factor, is defined as
$OR_i=e^{\beta}_i$

The regression coefficient $b_1$ from logistic regression is the estimated increase in the log odds of the dependent variable per unit increase in the value of the independent variable. In other words, the exponential function of the regression coefficients $(e^{b_1})$ in the OR associated with a one unit increase in the independent variable.