Ordinary least squares (OLS) regression is a method that allows us to find a line that best describes the relationship between one or more predictor variables and a response variable.

This method allows us to find the following equation:

**ŷ = b _{0} + b_{1}x**

where:

**ŷ**: The estimated response value**b**: The intercept of the regression line_{0}**b**: The slope of the regression line_{1}

This equation can help us understand the relationship between the predictor and response variable, and it can be used to predict the value of a response variable given the value of the predictor variable.

The following step-by-step example shows how to perform OLS regression in Python.

**Step 1: Create the Data**

For this example, we’ll create a dataset that contains the following two variables for 15 students:

- Total hours studied
- Exam score

We’ll perform OLS regression, using hours as the predictor variable and exam score as the response variable.

The following code shows how to create this fake dataset in pandas:

import pandas as pd #create DataFrame df = pd.DataFrame({'hours': [1, 2, 4, 5, 5, 6, 6, 7, 8, 10, 11, 11, 12, 12, 14], 'score': [64, 66, 76, 73, 74, 81, 83, 82, 80, 88, 84, 82, 91, 93, 89]}) #view DataFrame print(df) hours score 0 1 64 1 2 66 2 4 76 3 5 73 4 5 74 5 6 81 6 6 83 7 7 82 8 8 80 9 10 88 10 11 84 11 11 82 12 12 91 13 12 93 14 14 89

**Step 2: Perform OLS Regression**

Next, we can use functions from the statsmodels module to perform OLS regression, using **hours** as the predictor variable and score as the **response **variable:

import statsmodels.api as sm #define predictor and response variables y = df['score'] x = df['hours'] #add constant to predictor variables x = sm.add_constant(x) #fit linear regression model model = sm.OLS(y, x).fit() #view model summary print(model.summary()) OLS Regression Results ============================================================================== Dep. Variable: score R-squared: 0.831 Model: OLS Adj. R-squared: 0.818 Method: Least Squares F-statistic: 63.91 Date: Fri, 26 Aug 2022 Prob (F-statistic): 2.25e-06 Time: 10:42:24 Log-Likelihood: -39.594 No. Observations: 15 AIC: 83.19 Df Residuals: 13 BIC: 84.60 Df Model: 1 Covariance Type: nonrobust ============================================================================== coef std err t P>|t| [0.025 0.975] ------------------------------------------------------------------------------ const 65.3340 2.106 31.023 0.000 60.784 69.884 hours 1.9824 0.248 7.995 0.000 1.447 2.518 ============================================================================== Omnibus: 4.351 Durbin-Watson: 1.677 Prob(Omnibus): 0.114 Jarque-Bera (JB): 1.329 Skew: 0.092 Prob(JB): 0.515 Kurtosis: 1.554 Cond. No. 19.2 ==============================================================================

From the **coef** column we can see the regression coefficients and can write the following fitted regression equation is:

**Score = 65.334 + 1.9824*(hours)**

This means that each additional hour studied is associated with an average increase in exam score of **1.9824** points.

The intercept value of **65.334** tells us the average expected exam score for a student who studies zero hours.

We can also use this equation to find the expected exam score based on the number of hours that a student studies.

For example, a student who studies for 10 hours is expected to receive an exam score of **85.158**:

**Score = 65.334 + 1.9824*(10) = 85.158**

Here is how to interpret the rest of the model summary:

**P(>|t|):**This is the p-value associated with the model coefficients. Since the p-value for*hours*(0.000) is less than .05, we can say that there is a statistically significant association between*hours*and*score*.**R-squared:**This tells us the percentage of the variation in the exam scores can be explained by the number of hours studied. In this case,**83.1%**of the variation in scores can be explained hours studied.**F-statistic & p-value:**The F-statistic (**63.91**) and the corresponding p-value (**2.25e-06**) tell us the overall significance of the regression model, i.e. whether predictor variables in the model are useful for explaining the variation in the response variable. Since the p-value in this example is less than .05, our model is statistically significant and*hours*is deemed to be useful for explaining the variation in*score*.

**Step 3: Visualize the Line of Best Fit**

Lastly, we can use the **matplotlib** data visualization package to visualize the fitted regression line over the actual data points:

import matplotlib.pyplot as plt #find line of best fit a, b = np.polyfit(df['hours'], df['score'], 1) #add points to plot plt.scatter(df['hours'], df['score'], color='purple') #add line of best fit to plot plt.plot(df['hours'], a*df['hours']+b) #add fitted regression equation to plot plt.text(1, 90, 'y = ' + '{:.3f}'.format(b) + ' + {:.3f}'.format(a) + 'x', size=12) #add axis labels plt.xlabel('Hours Studied') plt.ylabel('Exam Score')

The purple points represent the actual data points and the blue line represents the fitted regression line.

We also used the **plt.text()** function to add the fitted regression equation to the top left corner of the plot.

From looking at the plot, it looks like the fitted regression line does a pretty good job of capturing the relationship between the **hours** variable and the **score** variable.

**Additional Resources**

The following tutorials explain how to perform other common tasks in Python:

How to Perform Logistic Regression in Python

How to Perform Exponential Regression in Python

How to Calculate AIC of Regression Models in Python

Hi, can you help me understand the statistics and methodology used in some research papers, need to apply heavy statistics to my research and am unable to navigate.

Hi Anuradha…I’d be happy to help you understand the statistics and methodology used in research papers. Let’s start with a general approach to breaking down the statistical methods and methodologies commonly found in research papers. If you have specific papers or methodologies in mind, please share those details, and we can delve into them further.

### General Approach to Understanding Research Papers

1. **Abstract and Introduction**

– **Purpose and Hypotheses**: Identify the research question, objectives, and hypotheses.

– **Background**: Understand the context and why the research is important.

2. **Literature Review**

– **Previous Research**: Note the key findings from previous studies and how they relate to the current study.

– **Gaps**: Identify the gaps in existing research that the current study aims to fill.

3. **Methodology**

– **Study Design**: Determine the type of study (e.g., experimental, observational, cross-sectional, longitudinal).

– **Sampling Methods**: Understand how the sample was selected (random, stratified, convenience) and the sample size.

– **Data Collection**: Note the tools and techniques used to collect data (surveys, experiments, observational methods).

– **Variables**: Identify the dependent and independent variables, along with any control variables.

4. **Statistical Analysis**

– **Descriptive Statistics**: Look for summaries of the data (means, medians, standard deviations, ranges).

– **Inferential Statistics**: Identify the statistical tests used (t-tests, ANOVA, regression analysis, chi-square tests).

– **Hypothesis Testing**: Understand the null and alternative hypotheses.

– **P-Values and Confidence Intervals**: Note the significance levels (typically p < 0.05) and the confidence intervals used. - **Effect Size**: Pay attention to the effect size, which indicates the magnitude of the findings. - **Assumptions**: Check if the assumptions of the statistical tests are mentioned and whether they were met. 5. **Results** - **Presentation of Findings**: Examine tables, graphs, and charts that present the data. - **Interpretation**: Understand how the authors interpret the statistical results in the context of their hypotheses. 6. **Discussion and Conclusion** - **Summary of Findings**: Note the main findings and how they relate to the hypotheses. - **Implications**: Understand the implications of the findings for the field. - **Limitations**: Identify the limitations acknowledged by the authors. - **Future Research**: Note any suggestions for future research. ### Common Statistical Methods 1. **Descriptive Statistics** - Mean, Median, Mode - Standard Deviation, Variance - Percentiles, Quartiles 2. **Inferential Statistics** - **Parametric Tests**: t-tests, ANOVA, regression analysis - Used when data meets certain assumptions (e.g., normality, homogeneity of variance). - **Non-Parametric Tests**: Mann-Whitney U, Kruskal-Wallis, chi-square tests - Used when data does not meet parametric assumptions. 3. **Regression Analysis** - **Linear Regression**: Analyzes the relationship between two continuous variables. - **Multiple Regression**: Examines the relationship between one continuous dependent variable and multiple independent variables. - **Logistic Regression**: Used for binary outcome variables. 4. **Correlation Analysis** - Pearson’s correlation coefficient - Spearman’s rank correlation coefficient 5. **Survival Analysis** - Kaplan-Meier estimator - Cox proportional hazards model ### Specific Help with Your Research If you provide details on: - The specific research paper or methodology you are working with. - The particular statistical methods or concepts you find challenging. - Any specific questions or issues you are encountering. I hope this helps!