Running an Analysis of Variance

Carrying on from the Hypothesis developed in Developing a Research Question I am trying to ascertain if there is a statistically significant relationship between the location and the sales price of a house in Ames Iowa. I have chosen to explore this in python. The tools used are pandas, numpy, and statsmodels.

Load in the data set and ensure the interested in variables are converted to numbers or categories where necessary. I decide to use ANOVA (Analysis of Variance) to test and TukeyHSD (Tukey Honest Significant Difference) for post-hoc testing my data set and my hypothesis.

This tells us that there are 25 neighbourhoods in the dataset.

We can create our ANOVA model with the smf.ols function and we will tilda SalePrice (dependent variable) with Neighborhood (independent variable) to build our model. We can then get the model fit using the fit function on the model and use the summary function to get our F-statistic and associated p value which we hope will be less than 0.05 so that we can reject our null hypothesis that there is no significant association between neighbourhood and sale price, therefore we can accept our alternate hypothesis that there is a significant relationship.

We get the output below which tells us that for 1460 observations with an F-statistic of 71.78 the p-value is 1.56e-225 meaning that the chance of this happening by chance is very very very small – 224 zero after the decimal point followed by 156, so we can safely reject the null hypothesis and accept the alternative hypothesis. Our adjusted R-squared is also .538 so our model is giving up a nearly 54% value for accuracy in including more than half of our training samples correctly. So our alternative hypothesis is that there IS a significant relationship between sale price and location (neighbourhood).

We know there is a significant relationship between neighbourhood and sale price but we don’t know which neighbourhood – remember we have 25 of these that can be different from eachother. So we must do some post-hoc testing. I will use the tukey hsd for this investigation

We can check the reject column below to see if we should reject any variations between neighbourhoods – but with 25 neighbourhoods, there are 25*24/2  = 300 relationships to check so there is a lot of output. Note we can output a box-plot to help visualise this too – see below the data for this output.

To visualise this we can use the pandas boxplot function although we probably have to tidy up the indices on the neighborhood (x) axis:

box_plot

Developing a Research Question

While trying to buy a house in Dublin I realised I had no way of knowing if I was paying a fair price for a house, if I was getting it for a great price, or if I was over-paying for the house. The data scientist in me would like to develop an algorithm, a hypothesis, a research question, so that my decisions are based on sound science and not on gut instinct. So for the last couple of weeks I have been developing algorithms to determine this fair price value. So my research questions is:
Is house sales price associated with socio-economic location?

I stumbled upon similar research by Dean DeCock from 2009 in his research determining the house price for Ames Iowa. So that is the data set that I will use. See the Kaggle page House Prices Advanced Regression Techniques to get the data.

I would like to study the association between the neighborhood (location) and the house price, to determine does location influence the sale price and is the difference in means between different locations significant.

This dataset has 79 independent variables with sale price being the dependent variable. Initially I am only focusing on one independent variable – the neighborhood, so I can reduce the dataset variables down to two, to simplify the computation my analysis of variance needs to perform.

Now that I have determined I am going to study location, I decide that I might further want to look at the bands of house size, not just the house size (square footage), but if I can turn those into categories of square footage, less than 1000, between 1000 and 1250 square feet, 1250 to 1500, > 1500 to see if there is a variance in the mean among these categories.

I can now take the above ground living space variable (square footage) and add it to my codebook. I will also add any other variables related to square footage for first floor, second floor, basement etc…

I then search google scholar, kaggle, dbs library for previous study in these areas, finding: a paper from 2001 discussing previous research in Dublin, however it was done in 2001 when a bubble was about to begin, and a big property crash in 2008 that was not conceived. http://www.sciencedirect.com/science/article/pii/S0264999300000407
Secondly Dean De Cock’s research on house prices in Iowa http://ww2.amstat.org/publications/jse/v19n3/decock.pdf

Based on my literature review I believe that there might be a statistically significant association between house location (neighborhood) and sales price. Secondary I believe there will be a statistically significant association between size bands (square footage band) and sales price. I further believe that might be an interaction effect between location & square footage bands and sales price which I would like to investigate too.

So I have developed three null hypotheses:
* There is NO association between location and sales price
* There is NO association between bands of square footage and sales price
* There is NO interaction effect in association between location, bands of square footage and sales price.

Running a LASSO Regression Analysis

A lasso regression analysis was conducted to identify a subset of variables from a pool of 79 categorical and quantitative predictor variables that best predicted a quantitative response variable measuring Ames Iowa house sale price. Categorical predictors included house type, neighbourhood, and zoning type to improve interpretability of the selected model with fewer predictors. Quantitative predictor variables include lot area, above ground living area, first floor area, second floor area. Scale were used for measuring number of bathrooms, number of bedrooms. All predictor variables were standardized to have a mean of zero and a standard deviation of one.

The data set was randomly split into a training set that included 70% of the observations (N=1022) and a test set that included 30% of the observations (N=438). The least angle regression algorithm with k=10 fold cross validation was used to estimate the lasso regression model in the training set, and the model was validated using the test set. The change in the cross validation average (mean) squared error at each step was used to identify the best subset of predictor variables.

Figure 1. Change in the validation mean square error at each step:

regression_coef_progmean_squared_error

Of the 33 predictor variables, 13 were retained in the selected model. During the estimation process, overall quality, above ground floor space, and garage cars being the main 3 variables. These 13 variables accounted for just over 77% of the variance in the training set, and performed even better at 81% on the test set of data.