Degrees of Freedom, Research Paper Example

Degrees of freedom are defined as the amount of variable values in the final calculation of a statistic. They are calculated by subtracting 1 from n, the total number of samples being measured. Inferential statistics allow one to examine data that is concrete knowledge and to extrapolate information from trends. Furthermore, it allows us to make assumptions about one population and then apply the information learned to a second group. The General Linear Model (GLM) is a line that can be used to plot single and multivariate regressions. It is helpful because it allows us to utilize our knowledge of statistical models in the formation of predictions concerning trends. Parametric and nonparametric statistics are used depending upon whether the data fits a normal curve. Parametric statistics are used when the data would fit a normal curve and nonparametric statistics are used when this is not the case. Parametric statistics rely more upon the mean of data, while nonparametric statistics rely upon the median more significantly.

If a p-value is equivalent to 0.05, it means that the results received will occur by chance only 5% of the time. Therefore, it is relatively reasonable that the trend that is being identified is real and not observed on the basis of random chance. A misconception about this p-value is that if the value is under it, then a phenomenon is definitely true. This is incorrect, because it is possible that an observed trend occurred because of chance even if it is statistically significant. Not all research should adhere to the p = 0.05 significance because some trends may be reasonably detected at other p-values (Fischer et al., 1938). For example, an experiment that measures an observation in which scientists cannot risk chance may have a p-value of 0.001, so that only 1% of observations were due to chance.

Effect size is necessary to be able to see a difference between two groups. If the effect size isn’t large, it is likely that the difference between the two groups will not be statistically significant (Stigler, 1986). Statistically significant results are valuable in the math world because it helps us determine which effects might be observed in real life. However, clinically relevant effects may not be statistically significant. It is nearly impossible to be certain of results based on statistical calculations, but these work as good starting points for a prediction.

Null Hypothesis Significance Testing (NHST) requires the scientist to create a null hypothesis that must be rejected in order to confirm that the alternative hypothesis has some validity (Lehmann, 1993). Major criticisms of this test include knowledge that these tests are only as good as the sample on which it is based. If the sample is not reflective of the target population, then we may inaccurately assume information about them. In addition, type I errors might occur, indicating that the conclusion may be wrong. Lastly, the presence of false positives and false negatives may interfere with the ability to draw a reasonable conclusion as well. Two proposed alternatives to this theory are Bayesian inference and Neyman-Pearson hypothesis testing. This first model is useful in significance testing as is the second.

The model that would be best to test the situation stated in activity 1 is Bayesian inference (Alrich, 1993). This is because it is difficult to determine the level of significance of the variables, which is useful to determine whether the results are meaningful in practical applications. Furthermore, Bayesian inferences can be used for both normal and non-normal data. Therefore, it would be more realistic to use this model for a variety of applications. It would be beneficial to support the model by using Neyman-Pearson hypothesis testing to confirm the results (Neyman et al., 1933). Since this test uses a likelihood ratio, it would be interesting to determine whether both statistical models share an agreement in terms of results. It would also be valuable to do this using other models to ensure that the data produced is reasonable.

References

Aldrich, J (2008). R. A. Fisher on Bayes and Bayes’ theorem. Bayesian Analysis, 3 (1): 161–170.

Fisher, R. A.; Yates, F. (1938). Statistical tables for biological, agricultural and medical research. London.

Lehmann, E. L. (1993). The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?. Journal of the American Statistical Association, 88 (424): 1242–1249.

Neyman, Jerzy; Pearson, Egon S. (1933). On the Problem of the Most Efficient Tests of Statistical Hypotheses.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 231 (694–706): 289–337.

Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Cambridge, Mass: Belknap Press of Harvard University Press.