Dataset csv free download kentucky derby

Additional Information Link opens in a new tab or window Explanation of points and adjustments. Time Period: Compiled by Raceday Added on Additional Information Link opens in a new tab or window Back to the Futures.

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Available data formats Google Sheet. Additional Information Link opens in a new tab or window Explanation of points and adjustments.

Time Period: Compiled by Raceday Added on Our data set derbyplus. We would like to use least squares linear regression techniques to model the speed of the winning horse as a function of track condition, field size, and trends over time. The first five and last five rows from our data set are illustrated in Table 1. Note that, in certain cases, we created new variables from existing ones:. With any statistical analysis, our first task is to explore the data, examining distributions of individual responses and predictors using graphical and numerical summaries, and beginning to discover relationships between variables.

This should always be done before any model fitting! We must understand our data thoroughly before doing anything else. First, we will examine the response variable and each potential covariate individually. Continuous variables can be summarized using histograms and statistics indicating center and spread; categorical variables can be summarized with tables and possibly bar charts.

Figure 1. Plot a shows winning speeds, while plot b shows the number of starters. In Figure 1. Plot b shows that the number of starters is mainly distributed between 5 and 20, with the largest number of races having between 15 and 20 starters.

The next step in an initial exploratory analysis is the examination of numerical and graphical summaries of relationships between model covariates and responses. The relationship between two continuous variables is depicted with scatterplots below the diagonal and correlation coefficients above the diagonal.

Here, we see that higher winning speeds are associated with more recent years, while the relationship between winning speed and number of starters is less clear cut. We also see a somewhat strong correlation between year and number of starters—we should be aware of highly correlated explanatory variables whose contributions might overlap too much. Relationships between categorical variables like track condition and continuous variables can be illustrated with side-by-side boxplots as in the top row, or with stacked histograms as in the first column.

As expected, we see evidence of higher speeds on fast tracks and also a tendency for recent years to have more fast conditions. These observed trends can be supported with summary statistics generated by subgroup. For instance, the mean speed under fast conditions is Finally, notice that the diagonal illustrates the distribution of individual variables, using density curves for continuous variables and a bar chart for categorical variables.

Trends observed in the last two diagonal entries match trends observed in Figure 1. By using shape or color or other attributes, we can incorporate the effect of a third or even fourth variable into the scatterplots of Figure 1. For example, in the coded scatterplot of Figure 1. Of course, any graphical analysis is exploratory, and any notable trends at this stage should be checked through formal modeling. At this point, a statistician begins to ask familiar questions such as:.

As you might expect, answers to these questions will arise from proper consideration of variability and properly identified statistical models. We will begin by modeling the winning speed as a function of time; for example, have winning speeds increased at a constant rate since ? Then, we might consider Model In the notation of this section,. Thus, according to our simple linear regression model, winning horses of the Kentucky Derby have an estimated winning speed of 2.

You may have noticed in Model 1 that the intercept has little meaning in context, since it estimates a winning speed in Year 0, when the first Kentucky Derby run at the current distance 1. One way to create more meaningful parameters is through centering. In this case, we could create a centered year variable by subtracting from each year for Model As Figure 1. We should also attempt to verify that our LINE linear regression model assumptions fit for Model 2 if we want to make inferential statements hypothesis tests or confidence intervals about parameters or predictions.

Most of these assumptions can be checked graphically using a set of residual plots as in Figure 1. In this case, the Residuals vs. Fitted plot indicates that a quadratic fit might be better than the linear fit of Model 2; other assumptions look reasonable. Leverage plot. Since no cutoff lines are even noticeable, there are no potential influential points of concern. We recommend relying on graphical evidence for identifying regression model assumption violations, looking for highly obvious violations of assumptions before trying corrective actions.

While some numerical tests have been devised for issues such as normality and influence, most of these tests are not very reliable, highly influenced by sample size and other factors. There is typically no residual plot, however, to evaluate the Independence assumption; evidence for lack of independence comes from knowing about the study design and methods of data collection.

In this case, with a new field of horses each year, the assumption of independence is pretty reasonable. Based on residual diagnostics, we should test Model 2Q, in which a quadratic term is added to the linear term in Model 2.

This model could suggest, for example, that the rate of increase in winning speeds is slowing down over time. In fact, there is evidence that the quadratic model improves upon the linear model see Figure 1. Fitted plot of Figure 1. We also may want to include track condition as an explanatory variable.

We could start by using fast as the lone predictor: Do winning speeds differ for fast and non-fast conditions? Since fast is numeric, we can use simple linear regression techniques to fit Model According to our fitted Model 3, the estimated winning speed under non-fast conditions is The answer is: we did!

Convince yourself that this is true, and that the equal variance assumption is needed. The beauty of the linear regression framework is that we can add explanatory variables in order to explain more variability in our response, obtain better and more precise predictions, and control for certain covariates while evaluating the effect of others.

For example, we could consider adding yearnew to Model 3, which has the indicator variable fast as its only predictor. In this way, we would estimate the difference between winning speeds under fast and non-fast conditions after accounting for the effect of time.

As we observed in Figure 1. A model with terms for both year and track condition will estimate the difference between winning speeds under fast and non-fast conditions for a fixed year ; for example, if it had rained in and turned the track muddy, how much would we have expected the winning speed to decrease?

Our new model estimates that winning speeds are, on average, 1. It appears our original model Model 3 may have overestimated the effect of fast conditions by conflating it with improvements over time. Through our new model, we also estimate that winning speeds increase by 0. This yearly effect is also smaller than the 0. So far we have been using linear regression for descriptive purposes, which is an important task.

We are often interested in issues of statistical inference as well—determining if effects are statistically significant, quantifying uncertainty in effect size estimates with confidence intervals, and quantifying uncertainty in model predictions with prediction intervals. Under LINE assumptions, all of these inferential tasks can be completed with the help of the t-distribution and estimated standard errors. In cases when model assumptions are shaky, one alternative approach to statistical inference is bootstrapping ; in fact, bootstrapping is a robust approach to statistical inference that we will use frequently throughout this book because of its power and flexibility.

Our primary assumption is that our original sample represents the larger population, and then we can learn about uncertainty in our parameter estimates through repeated samples with replacement from our original sample. If we wish to use bootstrapping to obtain confidence intervals for our coefficients in Model 4, we could follow these steps:. For example, the normal-theory confidence interval for the effect of fast tracks is 0. There are many variations on this bootstrap procedure.

For example, you could sample residuals rather than cases, or you could conduct a parametric bootstrap in which error terms are randomly chosen from a normal distribution. In addition, researchers have devised other ways of calculating confidence intervals besides the percentile method, including normality, studentized, and bias-corrected and accelerated methods Hesterberg ; Efron and Tibshirani ; Davison and Hinkley We will focus on case resampling and percentile confidence intervals for now for their understandability and wide applicability.

Adding terms to form a multiple linear regression model as we did in Model 4 is a very powerful modeling tool, allowing us to account for multiple sources of uncertainty and to obtain more precise estimates of effect sizes after accounting for the effect of important covariates. One limitation of Model 4, however, is that we must assume that the effect of track condition has been the same for years, or conversely that the yearly improvements in winning speeds are identical for all track conditions.

To expand our modeling capabilities to allow the effect of one predictor to change depending on levels of a second predictor, we need to consider interaction terms.

Amazingly, if we create a new variable by taking the product of yearnew and fast i. Interpretations of model coefficients are most easily seen by writing out separate equations for fast and non-fast track conditions:. In fact, using interaction allows us to model the relationships we noticed in Figure 1. Note that we interpret the coefficient for the interaction term by comparing slopes under fast and non-fast conditions; this produces a much more understandable interpretation for a reader than attempting to interpret the Modeling decisions should never be automated or made completely on the basis of statistical tests; subject area knowledge should always play a role in the modeling process.

Several tests and measures of model performance can be used when comparing different models for model building:. This model accounts for the slowing annual increases in winning speed with a negative quadratic term, adjusts for baseline differences stemming from track conditions, and suggests that, for a fixed year and track condition, a larger field is associated with slower winning times unlike the positive relationship we saw between speed and number of starters in our exploratory analyses.

The model explains We tested interaction terms for different effects of time or number of starters based on track condition, but we found no significant evidence of interactions. Having reviewed key ideas from multiple linear regression, you are now ready to extend those ideas, especially to handle non-normal responses and lack of independence.

This section provides a preview of the type of problems you will encounter in the book. Each of these examples appears later as an exercise, where you can play with the actual data or evaluate the analyses detailed in the articles. Roskes et al. Under time pressure, approach motivation leads to right-oriented bias.

Psychological Science [Online] 22 11 DOI: Abstract: Approach motivation, a focus on achieving positive outcomes, is related to relative left-hemispheric brain activation, which translates to a variety of right-oriented behavioral biases.