Linear regression models for predicting relationships between variables

Linear Regression models (Meier, Chapter 18 / 19)

These are used in order to determine whether a correlation (or relationship) exists between one element and another and, if so, in which direction (negative or positive).

The two variables are plotted on a graph. Independent variable on the x line (horizon); y- variable (dependent) on the vertical line. The pattern between them is called the 'slope'. The point where X and Y intersect online is called 'intercept'.

The theorem used tells us that the slope of the line will be equal to the change in x (IV) given changes of y (DV). The shape of the slope (their direction and gradient) describes the relationship between X and Y.

Linear regression, as are the previous models, is used apply results population sample to population as a whole. >Linear regression is also useful for predicting occurrences in that sphere. For instance, linear regression may be used to determine whether there is a correlation between vehicle collisions and rainy days. If so, one can predict that the stormier the weather the greater the quantity of collisions.

Goodness of Fit

We will want to know the amount of error i.e. how well the regression line fits the data. The distance a point is from the regression line is known as error. A certain calculation exists to find this out. Another goodness of fit measure is the standard error of the estimation where a calculation is used to find out the extent to which the results of the sampled population will correspond to the population as a whole. Thirdly, the coefficient of determination is used to measure the total variation in the independent variable (X). Complex calculations exist for this. (All of these calculations can be worked out by special computer programs too).

Linear regression has various assumptions:

1. For any value in X, the errors in predicting Y are normally distributed with a mean of zero.

2. Errors do not get larger as X becomes larger; rather the errors remain constant throughout slope regardless of the X value.

3. The errors of Y and X are independent of one another.

4. Both IV and DV (X and Y) must be interval variables (i.e. numerical data).

5. The relationships between X and Y are linear.

Ignoring these assumptions will result in faulty statistical conclusions.

Topic 2: Comparing 2 Groups

A researcher may run the same study on two different groups with one, for instance, acting as control and the other as experimental. He may then want to know whether differences are observed between the two groups.

1. Research and null hypothesis are drawn up stating that: (a) significant difference will be found, (b) significant difference will not be found between both groups.

e.g. Alternative Hyp. H1: Employees who have taken *program will have higher job scores

Null hyp (H0): There is no difference in scores between employees who have taken program and employees who have not.

2. Mean and standard deviation of each group is calculated

3. Standard error of each group is calculated

4. Aggregate standard error for both groups together is calculated

5. With all of those calculations, we can now work out t-score. Positing that resultant t-score is -.95. Looking that up in t-table and finding that it indicates probability of .1, we can now conclude that there is more than 1 chance in 10 that no difference (minus) will be found between the two groups.

The program, in other words, will have had insignificant impact.

c. T-test with dependent samples

In this case, the items are paired (e.g. with a pre -- and post test) therefore the differences between the scores of each of the individuals in both pre- and post- are calculated. Mean, SD, and SE are then calculated one of each of the differences. The t-score is calculated and probability assessed in order to see, for instance, whether differences have been found between pre- and post test.

Proportions

The t-test can also be used to investigate whether there are differences between two sample proportions.

Steinberg (2007) discusses the importance of the cause and effect relationship in policy research and the great role that it plays particularly in developing policies and in the policy process. He then goes on to describe the model of historical process tracing showing how it reveals causal mechanism at a degree and level of authenticity that statistical techniques cannot as readily or easily achieve. On the other hand, historical analysis is often confusing and more complex in its response showing -- as indeed is the case in real life -- that more than one explanation can be produced as explanation for causality. All of these factors may be necessary but none are exactly sufficient to describe the outcome response. Steinberg (2007) seeks to resolve the problem by suggesting that measurement criteria can be employed in order to rank the relative importance of the component causes. Nonetheless, when compared to linear regression, historical analyses is more complex in that it posts various factors as possible correlation. On the toehr hand, historical analyses seems more accurate in that it simulates reality by indicating that more than one element can be posited as explanatory factor for outcome.

The GAO case study paper in contrast focuses on qualitative, as opposed to quantitative research (thereby dramatically differing form Steinberg) and advises that GAO evaluators could better use caste methods in performing their works, proffering six applications of case study methods, explaining their similarities and differences, defining them, and determining their appropriateness.

The GAO case study, dealing with qualitative research, has little to do with the summary of research methods mentioned in this assignment. excepting the fact that it indicates that objective methodology could be used to reinforce the reliability, dependency, and consistency of results of the case studies. Taking this message to heart would lead one to employing some of the strategies mentioned in this assignment, and causing one to integrate quantitative with qualitative studies in that which is called triangulated experiments.