{"id":8649,"date":"2016-09-04T11:21:47","date_gmt":"2016-09-04T05:51:47","guid":{"rendered":"http:\/\/ucanalytics.com\/blogs\/?p=8649"},"modified":"2016-09-04T11:21:47","modified_gmt":"2016-09-04T05:51:47","slug":"bivariate-analysis-leverage-regression-case-study-example-part-3","status":"publish","type":"post","link":"https:\/\/ucanalytics.com\/blogs\/bivariate-analysis-leverage-regression-case-study-example-part-3\/","title":{"rendered":"Bivariate Analysis & Leverage – Regression Case Study Example (Part 3)"},"content":{"rendered":"
\n
\"Regression<\/a>

Leverage & Regression Case Study Example – by Roopam<\/p><\/div>\n

Welcome back to the\u00a0case study example for regression analysis where you are helping an investment firm make money through property price arbitrage. In the last two parts (Part 1<\/strong> <\/a>& Part 2<\/a><\/strong>) you started with the univariate analysis to identify patterns in the data including missing data and outliers. In the discussion section of the last part, Katya, Chetan, Abhishek and VC started an interesting discussion about the pros and cons of removal of missing data. I have a few opinions on missing data but let me reveal them later, for now I am really enjoying reading your ideas. Thanks, and please keep sharing your ideas.<\/p>\n

In this part you will further your investigation through\u00a0bivariate analysis. Bivariate analysis will eventually help you develop multivariate regression models in the latter\u00a0parts of this case study example. Through the bivariate analysis you will also identify how outliers can play havoc for your analysis. However, before that let’s discuss:<\/p>\n

Archimedes and Leverage in Regression<\/span><\/h2>\n

\"Regression<\/a>We have discussed Archimedes in an earlier article<\/a>\u00a0on YOU CANalytics. Yes, he is the famous Eureka!<\/em> guy. Archimedes was a mathematician, physicist, engineer, inventor, and astronomer. He used to say :\u00a0\u201c\u03a0\u0391 \u0392\u03a9 \u039a\u0391\u0399 \u03a7\u0391\u03a1\u0399\u03a3\u03a4\u0399\u03a9\u039d\u0399 \u03a4\u0391\u039d \u0393\u0391\u039d \u039a\u0399\u039d\u0397\u03a3\u03a9 \u03a0\u0391\u03a3\u0391\u039d<\/em>.\u201d OK if that’s Greek to you then you are right. The literal translation of this Greek sentence is :\u00a0\u201cGive me a place to stand and with a lever I will move the whole world.<\/em>\u201d.<\/p>\n

Archimedes in his famous quote was referring to the phenomenon called leverage. It is easy to experience leverage\u00a0if you try to open a door from the three different points (A, B, and C) displayed in the adjacent picture. You will notice that it requires much lesser effort or force to open the door the further you move away from the door hinges. This is the reason a door knob is placed as far away from the hinges to reduce your effort every time you open or close the door.<\/p>\n

Leverage plays an important role in regression models as we will notice in the next sections.<\/p>\n

Bivariate Analysis – Regression Case Study Example<\/span><\/h2>\n

Let’s come back to our case study example where you are a data science consultant for an investment firm. You are working with this\u00a0data set<\/a>\u00a0<\/strong>to estimate property price through regression modeling. In this part of the case study, you will do bivariate analysis between the numeric response\u00a0variable (house_price) and the remaining prospective predictor variables in this data set. The\u00a0bivariate analysis has different approaches based the nature of\u00a0predictor variables i.e. numeric or categorical. Before we continue with our analysis let’s revisit some core concepts of correlation analysis and scatter plots to analyse numeric predictor variables.<\/p>\n

\"Correlation<\/a>Correlation Analysis & Scatter Plot<\/span><\/h2>\n

One of the important measures while we perform bivariate analysis between 2 numeric variables is the correlation coefficient. Relation is the operative\u00a0word here. This coefficient represents relationship between the 2 variables. The range for correlation coefficient is between -1 and 1. The closer the correlation coefficient to 1 (i.e. 0.9 or 0.85) the higher positive relationship\u00a0the two variables have. Positive relationship means if the first variable grows than the second variable also grows. An example for positive correlation is age and height of teenagers.<\/p>\n

Similarly, the closer the correlation coefficient to \u00a0-1 (i.e -0.95 or – 0.8) the higher negative relationship the two variables have. As you must have guessed for the variables with negative correlation coefficient if the first variable grows than the second variable shrinks. An example for negative\u00a0correlation is your expenses and savings with a fixed income.<\/p>\n

Moreover, when the value of correlation coefficient is close to 0 (i.e. 0.01 or 0.05) then it represent there no or very little relationship between the 2 variables.<\/p>\n

In the next segment we will revisit our case study example and see how correlation coefficient\u00a0can be misleading in the presence of outliers. We will also connect this to leverage, the concept we discussed in the previous segment.<\/p>\n

Correlation and Leverage – Regression Case Study Example<\/span><\/h2>\n

Now let’s get back to the\u00a0dataset<\/a>\u00a0you are analyzing for your client. \u00a0You had\u00a0\u00a0started with\u00a0scatter plots of carpet area and house price with and without the outliers. You are particularly interested in\u00a0the correlation coefficients which you have placed at the top of these plots.<\/p>\n

\"Regression<\/a><\/p>\n

You noticed that the\u00a0extreme outlier (a mansion among the middle class houses) has a massive\u00a0leverage over the other observations. This has a huge impact on the correlation coefficient as well. Essentially, in the presence of this outlier you are drawing\u00a0the regression line between two dots on the scattered plot\u00a0: the extreme outlier and the whole bunch of data clubbed together (Exhibit 1). This is the reason the correction coefficient in this case is very close to perfect correlation i.e. 1. In Exhibit 2, when this outlier is removed the bunched up data looks like a more realistic scattered plot.<\/p>\n

Now, you want to\u00a0analyse all the numeric predictor variables against the response variable (house price) all at once. You will create scatter\u00a0plots and correlation coefficeints in the matrix format. Keep an eye for correlation coefficient and scatter plot for carpet area and house price you had already analysed in the above animation.<\/p>\n

R-code to create :\u00a0matrix plot with both scatter plot and correlation coefficient<\/a>.<\/em><\/p>\n

\"Regression<\/a><\/p>\n

You\u00a0have opted to remove the observation for the large mansion\u00a0from our dataset before the development of regression models. The matrix scattered plot without the outlier\u00a0looks like this.<\/p>\n

\"Regression<\/a><\/p>\n

Check out the histograms for all the numeric variables at the diagonal panels of this matrix plot. They all look well centered and nicely distributed. This\u00a0gives you a good confidence to move further with our analysis.<\/p>\n

If you look at the above matrix plot carefully you will also notice that distance between taxi stand, market, and hospital have significantly high correlations. Moreover, carpet and built-up area has almost perfect correlation. This indicates that in our data-set the build-up area is a derived field from the carper area because such a high correlation is almost impossible for a natural phenomenon. This is all pointing towards high correlation between predictor variables – the phenomenon is also known as multicollinearity. \u00a0We will explore more about multicollinearity in the next part when we will discuss principal component analysis. For now, let’s move further with bivariate analysis between categorical predictor variable and house price.<\/p>\n

Bivariate analysis – Categorical Predictor Variables<\/span><\/h2>\n

You have 2 categorical variables in our data-set: city category and parking availability. A good way to analyse categorical predictor variables\u00a0and numeric response variable is through a box plot. We can clearly see that there is a significant difference between the average price of houses based on the category of cities. The average prices are shown as in the middle of the boxes. R code :\u00a0bivariate analysis – categorical variable<\/a>.<\/p>\n

\"Bivariate<\/a><\/p>\n

Moreover to validate what you see in the box plot, you have performed pair-wise t test for each category. The results for pair-wise t-test shown at the top of the box plot in red. You noticed that\u00a0P(A=B)~0 means that there are almost 0% chances that average price of houses in cat A city is equal to cat B city.\u00a0Go ahead and plot a similar chart for parking and house prices. Also, let us know what you see.<\/p>\n

Sign-off Note<\/span><\/h4>\n

Archimedes did not find an enormously long lever and a place far-far away to move the earth. However, he did shake the earth with his ideas and had a leverage over the scientific thinking.<\/p>\n

In the next article, we will progress with multivariate regression model. The bivariate analysis in this part has already offered clues about the structure of our final model.<\/p>\n","protected":false},"excerpt":{"rendered":"

Welcome back to the\u00a0case study example for regression analysis where you are helping an investment firm make money through property price arbitrage. In the last two parts (Part 1 & Part 2) you started with the univariate analysis to identify patterns in the data including missing data and outliers. In the discussion section of the<\/p>\n

Read More…<\/a><\/p>\n","protected":false},"author":1,"featured_media":8654,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_newsletter_tier_id":0,"jetpack_publicize_message":"","jetpack_is_tweetstorm":false,"jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[80],"tags":[],"jetpack_publicize_connections":[],"yoast_head":"\nBivariate Analysis & Leverage - Regression Case Study Example (Part 3) – YOU CANalytics |<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ucanalytics.com\/blogs\/bivariate-analysis-leverage-regression-case-study-example-part-3\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bivariate Analysis & Leverage - Regression Case Study Example (Part 3) – YOU CANalytics |\" \/>\n<meta property=\"og:description\" content=\"Welcome back to the\u00a0case study example for regression analysis where you are helping an investment firm make money through property price arbitrage. 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Welcome to a new data science case study example on YOU CANalytics to identify the right housing price. Pricing is a highly important and\u00a0specialized function for any business. A right price\u2026","rel":"","context":"In "Pricing Case Study Example"","block_context":{"text":"Pricing Case Study Example","link":"https:\/\/ucanalytics.com\/blogs\/category\/pricing-case-study-example\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/07\/Connect-the-Dots.jpg?fit=397%2C603&ssl=1&resize=350%2C200","width":350,"height":200},"classes":[]},{"id":8488,"url":"https:\/\/ucanalytics.com\/blogs\/data-preparation-regression-pricing-case-study-example-part-2\/","url_meta":{"origin":8649,"position":1},"title":"Data Preparation for Regression – Pricing Case Study Example (Part 2)","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"In the last post we had started a case study example for regression analysis to help an investment firm make money through property price arbitrage\u00a0(read part 1 :\u00a0regression case study example).\u00a0This is an interactive case study example and required your help to move forward. These are some of your observations\u2026","rel":"","context":"In "Analytics Labs"","block_context":{"text":"Analytics Labs","link":"https:\/\/ucanalytics.com\/blogs\/category\/analytics-labs\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/08\/Regression-analysis.jpg?fit=448%2C528&ssl=1&resize=350%2C200","width":350,"height":200},"classes":[]},{"id":9018,"url":"https:\/\/ucanalytics.com\/blogs\/step-step-regression-models-pricing-case-study-example-part-5\/","url_meta":{"origin":8649,"position":2},"title":"Step by Step Regression Modeling Using Principal Component Analysis – Case Study Example (Part 5)","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"This is a continuation of our case study example to estimate property pricing. In this part, you will learn nuances of regression modeling by building three different regression models and compare their results.\u00a0We will also use results of the principal component analysis, discussed in the last part, to develop a\u2026","rel":"","context":"In "Pricing Case Study Example"","block_context":{"text":"Pricing Case Study Example","link":"https:\/\/ucanalytics.com\/blogs\/category\/pricing-case-study-example\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/09\/Sumo-and-Regression-Model.jpg?fit=918%2C384&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/09\/Sumo-and-Regression-Model.jpg?fit=918%2C384&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/09\/Sumo-and-Regression-Model.jpg?fit=918%2C384&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2016\/09\/Sumo-and-Regression-Model.jpg?fit=918%2C384&ssl=1&resize=700%2C400 2x"},"classes":[]},{"id":5782,"url":"https:\/\/ucanalytics.com\/blogs\/how-effective-is-my-marketing-budget-regression-with-arima-errors-arimax-case-study-example-part-5\/","url_meta":{"origin":8649,"position":3},"title":"How Effective is My Marketing Budget? – Regression with ARIMA Errors, Case Study Example (Part 5)","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"So far we have covered the following topics in this case study example\u00a0on time series forecasting and ARIMA models: Part 1\u00a0: Introduction to time series modeling & forecasting Part 2: Time series decomposition to decipher patterns and trends before forecasting Part 3: Introduction to ARIMA models for forecasting Part 4:\u2026","rel":"","context":"In "Manufacturing Case Study Example"","block_context":{"text":"Manufacturing Case Study Example","link":"https:\/\/ucanalytics.com\/blogs\/category\/manufacturing-case-study-example\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2015\/07\/rope-walk.jpg?fit=480%2C640&ssl=1&resize=350%2C200","width":350,"height":200},"classes":[]},{"id":3973,"url":"https:\/\/ucanalytics.com\/blogs\/model-selection-retail-case-study-example-part-7\/","url_meta":{"origin":8649,"position":4},"title":"Model Selection – Retail Case Study Example (Part 7)","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"Model Selection This is a continuation of our retail case study example for campaign and marketing analytics. In the previous two parts, we discussed a couple of decision tree algorithms (CART and C4.5)\u00a0for classification. Recall a previous case study example on\u00a0banking and risk management where we discussed logistic regression\u00a0which is\u2026","rel":"","context":"In "Marketing Analytics"","block_context":{"text":"Marketing Analytics","link":"https:\/\/ucanalytics.com\/blogs\/category\/marketing-analytics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2014\/09\/photo.jpg?fit=1200%2C1029&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2014\/09\/photo.jpg?fit=1200%2C1029&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2014\/09\/photo.jpg?fit=1200%2C1029&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2014\/09\/photo.jpg?fit=1200%2C1029&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2014\/09\/photo.jpg?fit=1200%2C1029&ssl=1&resize=1050%2C600 3x"},"classes":[]},{"id":8700,"url":"https:\/\/ucanalytics.com\/blogs\/principal-component-analysis-step-step-guide-r-regression-case-study-example-part-4\/","url_meta":{"origin":8649,"position":5},"title":"Principal Component Analysis: Step-by-Step Guide using R- Regression Case Study Example (Part 4)","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"Principal component analysis is a wonderful technique for data reduction without losing critical information. Yes, you could reduce the size of 2GB data to a few MBs without losing a lot of information. This is like a mp3 version of music. 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