Inferential Statistics: Descriptive statistics deals with the analysis of data collected on the sample (subjects). Inferential statistics is – in contrast – concerned with what the subjects can tell us about the larger population they represent. It is usually impossible to observe the entire population of our interest. We resort to observing and measuring the characteristics of a sample of that population instead. Example: To study the software defect discovery capability of inspectors, we cannot observe everyone who has and will do inspections. Instead we look at a sample population.

The sample is selected and used as if it represents the general population. Therefore, it better do so! Strict sampling procedures must be followed. At any rate, due to natural (chance) variability between any sample and its general population, we CANNOT say, with certainty that any observation made with respect to the sample would extend to the general population. We can only talk about the probability that it would. Example: In a test designed to find the extent to which programmers with no training in testing would find program defects compared to those

that have had such training, we find that the sample of programmers under our study found on average 5.3 defects in an hour of testing. The control group (the ones who were trained in testing) found on average 5.8 defects. What does this finding mean? Given the difference between the sample groups, does this mean that there is a difference between the trained population and non-trained population? Is this difference sufficiently significant to be 1) not just by chance, 2) or small enough to not signify a real difference? (so for example to recommend discontinuing tester training.)

Here, we are testing the null hypothesis that: There is no real difference between the two population means. We know about the null hypothesis and the type I and type II errors. To go further, however, we need to distinguish between a population parameter and a sample statistic. A sample statistic is a statistic describing the sample drawn. For example, taking the mean of the data-points from the sample. A population parameter is a characteristic of the entire population. For example if we take the mean of ALL data-points in the population.

Our task is usually to estimate population parameters and to draw conclusions about population parameters based on sample statistics. Inferential statistics are used to compute the probability of obtaining the observed data if the null-hypothesis were true. If the probability is small, then it is unlikely that the null-hypothesis is true. The somewhat arbitrary cut-off points (called alpha levels) were introduced to cater for such measure.

Testing for mean differences: Inferential statistics are used most frequently to evaluate mean differences between groups. We can use such techniques to specify the research hypothesis in terms of mean differences. There are a number of tests for evaluating mean differences in two or more groups. These include the: Simple t-test Correlated t-test Analysis of Variance (ANOVA)

Simple t-test: This test is typically used with score data from two independent samples of subjects (e.g. trained and non-trained). The null-hypothesis is that there is NO difference in the two population means. In other words , the observed difference between the sample means is due only to chance. The test statistic is called the t statistic. We compute the t statistic and the probability (p value) of obtaining this t value if the null hypothesis is true. If p is less than our alpha level, we reject the null-hypothesis and conclude that the population means are different.

We compute the t statistic using the following formula: and are the means of the two samples and and are the sum of squares of each sample.

Correlated t-test: When we do not have independent samples, we use the correlated t-test. Examples of such design might be within-subject design, where the same subjects appear in each group or matched-subjects design, where all subjects are paired and then randomly assigned so that one member of the pair goes into one group and the other into another group.

An example of within-subject design might be when a group of subjects are given an error free, a somewhat buggy and a bug-riddled program to test. The testing time of each person testing each program is recorded. As each subject is exposed to all levels of the independent variable, the scores in each condition are correlated with the scores in the other condition (i.e. performance in one is correlated with performance in the other condition.) The critical comparison is the difference between correlated groups on the dependent variable (rate of defect identification).

An example of matched subject design would be when comparing the effectiveness of training in two different methods of testing software. If tester 1 is trained in method A he or she cannot be trained in method B. We need a new subject, say subject 2 who is the paired member of subject 1. Now we can compare the two groups using a correlated t-test.

Analysis of Variance (ANOVA): When we have more than two groups and want to test for mutual mean difference between all the groups we usually use a form of the ANOVA method. The term is actually confusing as the test compares the means of the various groups but it does so by computing and comparing the different population variance estimates. One advantage of ANOVA is that it allows analysis of one,two or several independent variables at the same time. Each independent variable is called a factor and the research design with more than one factor is called factorial research design.

We refer to a study with only one independent variable as a one-way ANOVA, with two factors, it is called two-way ANOVA and so forth. Example: In a study of tester behavior it was found that testers do generally better when there is a moderate level of other activity in the work-place. Testing in total silence usually ends in poorer result. Further investigation found that it is not the absence of ambient noise that results in poor performance alone but that conditions such as the mood of the tester, the load under which they work, personal problems, etc. are also contributing factors. It seems therefore that silence is a necessary but insufficient condition for degradation of performance as are moods, problems and loads. We can now set up a factorial design to test two or

several independent variables may be considered several independent variables may be considered. Let us say we concentrate on two level of ambient noise (binary; relatively noisy or silent) and mood (negative or positive). So we have two factors and two levels for each. This gives four treatment combinations leading to a 2 by 2 factorial design.

Multivariate Analysis of Variance (MANOVA): The difference between MANOVA and ANOVA is in the dependent variable. In ANOVA we can have multiple independent variables but one dependent variable. In MANOVA we can have multiple dependent variables also.

Mann-Whitney U test: This is the comparison inferential test to be used if we have two ordinal groups of data and the groups are independent. Wilcoxon signed-rank test: This is the comparison inferential test to be used if we have two ordinal groups of data and the groups are not independent.

Kruskal-Wallis one-way ANOVA: This is the comparison inferential test to be used if we have more than two ordinal groups of data and the groups are independent. Friedman two-way ANOVA: This is the comparison inferential test to be used if we have more than two ordinal groups of data and the groups are not independent.

Chi-square goodness of fit: This is when we have one group being compare against a hypothetical situation or a theory. Chi-square test for independence: This is when we have more than one group being compared against each other.

Selecting appropriate Statistical Analysis procedures. Flowchart 0 Start Go to Flowchart 1 Go to flowchart 2 Start with 1st hypothesis Nominal Type of data for that dependent variable? What is the dependent variable? Go to flowchart 3 Ordinal Score Go to flowchart 4 yes Are there more research hypotheses to evaluate? Selecting appropriate Statistical Analysis procedures. Flowchart 0 End No

Flowchart 1: Start-up Start Identify the following: Level of constraint (Naturalistic, case-study, correlational, …, experimental) Independent variables Levels and scale of independent variables Each independent variable manipulated or not. Type of design (independent groups, correlated groups, mixed..) Dependent variables Levels and scale of dependent variables Operational procedures Research hypothesis Type of test needed 9relationship or difference) End

Flowchart 2: Nominal Data Infer. Start Frequency count Comparing one group against hypothetical situation or two or more groups Desc. Inferential or Descriptive one Chi-square goodness-of-fit Back to flowchart 0. More hypotheses? Two+ Chi-square test of independence Flowchart 2: Nominal Data

Flowchart 3: Ordinal Data Start Median central Variab. Desc. Type of description Infer. Inferential or Descriptive Range Number of groups Three+ relation Groups independent or correlated Two Corr.. Spearman rank correlation Groups independent or correlated Indep. Freiedman two way ANOVA Corr. Wilcoxon signed-rank test Kruskal-Wallis one way ANOVA Back to flowchart 0. More hypotheses? Indep. Mann-Wittney U-test Flowchart 3: Ordinal Data

Flowchart 4: Score Data Number of factors Start Mean, median,mode,Z Two+ one central Variab. Desc. Type of description Infer. Inferential or Descriptive Variance, std one Number of groups Two+ relation two Pearson product moment Groups independent or correlated Corr. t test for correlated groups ANOVA Back to flowchart 0. More hypotheses? Indep. t test for independent groups Single group t test