The Degrees of Freedom

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The degrees of freedom (df) or several degrees of freedom refers to the number of observations in a sample minus the number of (population) parameters being estimated from the sample data. All this means that the degrees of freedom are a function of both sample size and the number of independent variables. In other words, it is the number of independent observations out of a total of ($n$) observations.

Degrees of Freedom

In statistics, the degrees of freedom are considered as the number of values in a study that is free to vary. Degree of freedom example in real life; if you have to take ten different courses to graduate, and only ten different courses are offered, then you have nine degrees of freedom. Nine semesters you will be able to choose which class to take; the tenth semester, there will only be one class left to take – there is no choice, if you want to graduate, this is the concept of the degrees of freedom (df) in statistics.

Let a random sample of size $n$ be taken from a population with an unknown mean $\overline{X}$. The sum of the deviations from their means is always equal to zero i.e.$\sum_{i=1}^n (X_i-\overline{X})=0$. This requires a constraint on each deviation $X_i-\overline{X}$ used when calculating the variance.

\[S^2 =\frac{\sum_{i=1}^n (X_i-\overline{X})^2 }{n-1}\]

This constraint (restriction) implies that $n-1$ deviations completely determine the nth deviation. The $n$ deviations (and also the sum of their squares and the variance in the $S^2$ of the sample) therefore $n-1$ degrees of freedom.

A common way to think of df is the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a parameter of the population from which that sample is drawn. For example, if we have two observations, when calculating the mean we have two independent observations; however, when calculating the variance, we have only one independent observation, since the two observations are equally distant from the mean.

Degrees of Freedom

Single sample: For $n$ observation one parameter (mean) needs to be estimated, which leaves $n-1$ degree of freedom for estimating variability (dispersion).

Two samples: There are a total of $n_1+n_2$ observations ($n_1$ for group1 and $n_2$ for group2) and two means need to be estimated, which leaves $n_1+n_2-2$ degree of freedom for estimating variability.

Regression with p predictors: There are $n$ observations with $p+1$ parameters that need to be estimated (regression coefficient for each predictor and the intercept). This leaves $n-p-1$ degrees of freedom of error, which accounts for the error degrees of freedom in the ANOVA table.

Several commonly encountered statistical distributions (Student’s t, Chi-Squared, F) have parameters that are commonly referred to as degrees of freedom. This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector. If $X_i; i=1,2,\cdots, n$ are independent normal $(\mu, \sigma^2)$ random variables, the statistic (formula) is $\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{\sigma^2}$, follows a chi-squared distribution with $n-1$ degree of freedom. Here, the degree of freedom arises from the residual sum of squares in the numerator and in turn the $n-1$ degree of freedom of the underlying residual vector $X_i-\overline{X}$.

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Binary Logistic Regression Minitab Tutorial

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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 the presence or absence of a particular disease, this kind of variable is known as a dichotomous variable i.e. binary in nature).

Binary Logistic Regression

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 the model sub-dialog box.

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

Minitab Tutorial for Binary Logistic Regression

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.

Step 1:  Choose Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model.

Binary Logistic Regression Minitab Tutorial

Step 2:  Do one of the following:

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

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

If you have summarized data, then follow these steps:

Binary Logistic Regression 3
  1. Choose Response in event/trial format, from the combo box on top of the dialog box.
  2. In the 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 the Number of trials, enter the column that contains the corresponding number of trials.

Step 4:  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.

Step 5:  If you like, use one or more of the dialog box options, then click OK.

Minitab Binary Logistic Regression Options

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

  • The 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 the type of raw material, that explain changes in the response. The predictor is also called the X variable.

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

Binary Logistic Regression Minitab Tutorial

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Multiple Regression Model Introduction (2015)

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Introduction to Multiple Regression Model

A multiple regression model (a regression having multi-variable) is referred to as a regression model having more than one predictor (independent and explanatory variable) to explain a response (dependent) variable. We know that simple regression models have one predictor used to explain a single response while for the 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 is not based on predictors or the addition of more predictors in the simple regression model, it is referred to as the parameter of variability (parameters attached with predictors). If the parameters of variability have a 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 linear, though this assumption can never be confirmed in the case of multiple linear regression.

However, as a rule, it is better to look at a bivariate scatter diagram of the variable of interest, you check that there should be no curvature in the relationship. A scatter matrix plot is a more useful visualization between variables of interest.

The multiple regression model also allows us 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.

General Form of Multiple Regression Model

A multiple regression model has the form

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

Here $y$ is continuous variables and $x$’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.

Multiple Regression Model

Example of Multiple Regression Model

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

  • How rainfall, temperature, and amount of fertilizer impact and affect crop growth
  • Influence of various factors (such as cholesterol, blood pressure, or diabetes) on health outcomes
  • Blood pressure depends on variables, for example, gender, age, height, weight, exercise, diet, and medication.
  • The Weight of a person is linearly related to their height and age.
  • Studying the effect of education, gender, and profession on income.
  • The price of a house depends on the size of the house, number of rooms, community, facilities available, etc.

Assumptions of the Multiple Regression Model

Multiple regression models also have some assumptions that need to be followed or fulfilled. For example, the residuals should be normally distributed. There should be no collinearity/ multicollinearity among the regressors/ independent variables. The variance of error terms should be homoscedastic, and error terms should be not correlated (no autocorrelation).

Common Applications of Multiple Regression Models

  • Marketing: Predicting customer spending based on factors like income, gender, age, and advertising exposure.
  • Social Science: Analyzing the factors that influence voting behavior, such as gender, education level, income, and political party affiliation.
  • Finance: Estimating stock prices based on company earnings, economic indicators, and market trends.
  • Predicting house prices: One can use factors like square area, number of bedrooms, and location to predict the selling price of a house.
  • Identifying risk factors for diseases: Researchers can use multiple regression to see how lifestyle choices, genetics, and environmental factors contribute to the risk of developing a particular disease.

Read Assumptions of Multiple Regression Model

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Logistic regression Introduction (2015)

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Logistic regression was introduced in the 1930s by Ronald Fisher and Frank Yates and was first proposed in the 1970s as an alternative technique to overcome the limitations of ordinary least square regression in handling dichotomous outcomes. It is a type of probabilistic statistical classification model which is a non-linear regression model, and can be converted into a 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 a value as zero or one i.e., dichotomous variable.

Logistic Regression Model

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.

Logistic Regression

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 a linear model. As $\pi$ is the probability of the events in which we are interested if we take the ratio of the probability of success and failure then the model becomes a linear model.

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

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

Binary Logistic Regression in Minitab

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