Acorn Size and Oak Tree Range

Make a scatter plot of tree range vs. acorn size. Find the correlation. Does there appear to be any linear relation between range and acorn size?

• Be able to use your software to make a scatterplot.
• Be able to interpret the display (for example, recognize patterns or spotting outliers).
• Be able to use software to calculate the correlation coefficient. Be able to interpret this value.

The correlation is 0.076. There does not seem to be much of a linear relation at all.

a) Examine the summary statistics for tree range. What are the mean and the standard deviation? What do these values tell you about the likely shape of the distribution?

b) Now regress Range on Acorn size. If you want to be able to predict tree range by knowing acorn size, how much does this regression equation help you? Explain.

• a) Be able to calculate (using software where appropriate) and interpret the principle measures of center, relative standing, and spread (rates, percentages, mean, median, mode, percentiles, lower quartile, upper quartile, minimum, maximum, standard score, the five-number summary, variance, and standard deviation).
• a) Understand the relationship between key statistical summaries and histograms (For example, mean, median, standard deviation, variance, and quartiles).
• b) Be able to use the computer to calculate simple linear regression estimates and know how to interpret the resulting output including (y = response variable,= estimated response, and x = the explanatory variable): the R-squared value (the square of the correlation measures the proportion of the variance of one variable that can be explained by straight-line dependence on the other variable.)
  

a) The mean of range is 788256 km², the SD is 805482 km². Since the SD is larger than the mean, we would expect the distribution for oak tree geographical range to have a long right tail.

b)

This regression equation does not help at all. The R² value is 0.6%. In other words, only 0.6% of variability in Range is explained by the regression on Acorn size. This is essentially zero (or equivalently, the rms error is no smaller than the standard deviation of the Range variable). Thus our predicted value of Range will not be any closer to the actual value with the regression equation than without it.

Examine a normal probability plot of tree range and a normal probability plot of acorn size. Do these data appear to follow the normal distribution? If not, what shape do they have? You may wish to look at the histograms of the variables.

• Be able to use a normal quantile plot to assess the normality of a given dataset.

        
The normal probability plots indicate that the distributions of tree range and acorn size, respectively, have a longer than normal right tail. The histograms below confirm this idea.
        

The authors suggest a transformation of the data. Based on what you learned from Question 3, which transformation would you suggest? 

• Be able to reason what transformations would be appropriate in a real problem.

Since both distributions have a longer than normal right-tail, a log transformation might be suggested to dampen the influence of the largest observations. Also, relationships between the transformed variables would then be interpretable on a relative scale.

a) Transform the data using the log transformation on both the range and the size. Now make a scatter plot of Ln(range) vs. Ln(acorn size). How did the correlation change? Does the correlation surprise you? Do you see any obvious reason that might help explain the correlation?

b) Examine the normal probability plots of Ln(range) and Ln(acorn size). What do they tell you?

• a) Be able to use your software to make a scatterplot. Be able to interpret the display (for example, recognize patterns or spotting outliers).
• a) Be able to interpret results pertaining to transformed data.
• b) Be able to use a normal quantile plot to assess the normality of a given data set.

 

a)
     
The correlation is about the same, 0.079. The transformation has changed the shape of the cloud of points to show a more linear pattern, but the outlier in the bottom right hand corner of the scatter plot accounts for the low correlation.

b)
     
The normal probability plot of Ln(Acorn size) indicates approximate normality of this variable. The normal probability plot of Ln(Range) shows the presence of an extreme outlier.

Compare boxplots of tree range by geographical region in order to investigate the relationship between tree range and region. What do you learn?

• Be able to make and interpret a boxplot.
• Be able to use the IQR rule for identifying outliers in a boxplot.

 

      
The comparative boxplots show that the bigger ranges are found in the Atlantic region, the smaller ranges are found in the California region.

a) Make a scatter plot of Ln(range) vs. Ln(acorn size) for the Atlantic region. Is the correlation any better than that found in Question 1?

b) Make a scatter plot of Ln(range) vs. Ln(acorn size) for the California region. What is the correlation? Why do you think that the correlation is so low?

• a) Be able to use your software to make a scatterplot.
• a) Be able to interpret the display (for example, recognize patterns or spotting outliers.)
• b) Be able to use software to calculate the correlation coefficient.
• b) Be able to interpret this value.

a) The scatter plot for the Atlantic Region shows a moderate positive association. The correlation has improved to 0.624.
      

b) The correlation is only 0.0203 for the California region. This low correlation is due to the presence of the outlier at the bottom right of the plot. The rest of the data seem to be associated in a stronger manner.
      

a) Examine the residuals from the regression of Ln(range) on Ln(acorn size) and the indicator for region. Does there seem to be an outlier? Can you identify the outlier?

b) What is unusual about the species of oak represented by this outlier? (Hint: Consult the map of the region.) How does this information help you understand this data point and explain the outlier?

• a) Understand that simple linear regression is inappropriate when an outlier (or influential observation) drives the results.
• b) Understand how to check the key model assumptions that are made in regression using residual plots and other model diagnostics.

a) A scatter plot of the residuals versus predicted values is shown below with the Atlantic region denoted by red •'s and the California region denoted by blue x's.
          
This plot illustrates the importance of incorporating information about the region into the prediction. Also, there does appear to be an outlier. The outlier represents Quercus tomentella Engelm, which is the species that only grows on the Channel Islands and the island of Guadalupe.

b) It is the only species of oak in the data set which does not grow on the continent. Its geographic range is limited by the size of the islands (about 1300 sq. km). This makes this species different from the other species in the data set. Since the range is limited yet the size of the acorn is not, this oak tree does not follow the same pattern as the others.