Lab 4 Regression analysis

1. Multivariable matrix
1. Use the multi-variable chart function of StatPlus to create a matrix of scatterplots with temperature, salinity, density (sigma_t), AOU, and CO₂. (This is not mandatory for the lab results so if your computer can not handle the extra memory it takes up this step can be skipped.)
2. Use the multivariate function of StatPlus to create a matrix of correlation coefficients with temperature, salinity, density (sigma_t), AOU, and CO₂.
3. Which variables are significantly correlated concentrations of CO₂?
4. What does this imply are the dominating mechanisms controlling changes in CO₂?
5. Are these the only factors that control the concentration of CO₂?
2. Calculate the regression equations:
1. Use variance and covariance  to calculate Least squares method for the variable that is best correlated with CO₂. (Remember that VAR and COVAR function in excel will calculate the variance and covariance; however, COVAR must be multiplied by n and divided by (n-1))
2. Use REGRESSION function in excel for the variable that is best correlated with CO₂.
3. Use VAR and COVAR function in excel to calculate Reduced major axis for the variable that is best correlated with CO₂.
4. Plot each of the above regression equations on a plot.
5. For each of the above equations calculate the sum of squared residuals.
6. Which method shows the smallest sum of least squares residuals and thus is the sum of least squares method?
7. Which regression equation is most appropriate if we assume that the measurement errors for both CO₂ and the variable best correlated with CO₂ and why?
3. Multivariable predictions
1. Plot the residuals from the predicted CO₂ and the measured CO₂ verses the variable that is also well correlated (the second largest R² not including oxygen) with CO₂.
2. Is there a trend? If so, use the linear trend to create an equation which will predict CO₂ based two other variables and calculate a sum of least squares residual and subtract that from the sum of squares of CO₂ (var(CO₂)*(n-1))? What fraction of variance in TCO2 does this new equation represent.
3. Assuming no error in the measurement of the two variables that best correlate with CO₂ use the LINEST or REGRESSION function to calculate a multi variable least squares fit to the TCO2 data. As above, calculate the sum of squared residuals and calculate how much of the variance in CO₂ can be explained using least squared fit with multiple (two) parameters.
4. Perform a multi variate regression to predict the CO₂ with all the variables ( temperature, salinity, density (sigma_t), Dt and AOU) using the regression function. Does this improve the sum of squared residuals?
4. Interpretation of the data:
1. Using the results above explain as precisely as possible what they imply about the variability of CO₂ as observed by flip. You should format this in the following order:

1. Results: Explain exactly what your statistical analysis shows (by explaining graphs and calculations produced above). It will also be helpful to show a time series of CO₂ and the two parameters that best correlate with CO₂ to set the stage.
2. Summary: Explain what the two parameters that best correlate with CO₂ imply about the mechanisms that result in variability of CO₂. Explain which process seems to dominate and how this is different from the basic processes that I have explained in the introduction to this lab. In particular, you should note the photosynthetic quotient that we expect (moles O₂ produced/ moles of CO₂ used- see experiment section of the introduction to this lab).