4 Inferential techniques

Inferential techniques involve generalizing from a sample to the whole population. It also involves testing a hypothesis. A hypothesis must be stated in mathematical/statistical terms that make it possible to calculate the probability of possible samples assuming the hypothesis is correct. Then a test statistic must be chosen that will summarize the information in the sample that is relevant to the hypothesis. A null hypothesis is a hypothesis that is presumed true until a hypothesis test indicates otherwise. Typically it is a statement about a parameter that is a property of a population. The parameter is often a mean or a standard deviation.

Not unusually, such a hypothesis states that the parameters, or mathematical characteristics, of two or more populations are identical. For example, if we want to compare the test scores of two random samples of men and women, the null hypothesis would be that the mean score in the male population from which the first sample was drawn, was the same as the mean score in the female population from which the second sample was drawn:

where:

H₀ = the null hypothesis

μ₁ = the mean of population 1, and

μ₂ = the mean of population 2.

The equality operator makes this a two-tailed test. The alternative hypothesis can be either greater than or less than the null hypothesis. In a one-tailed test, the operator is an inequality, and the alternative hypothesis has directionality:

These are sometimes called a hypothesis of significant difference because you are testing the difference between two groups with respect to one variable.

Alternatively, the null hypothesis can postulate that the two samples are drawn from the same population:

A hypothesis of association is where there is one population, but two traits being measured. It is a test of association of two traits within one group.

The distribution of the test statistic is used to calculate the probability sets of possible values (usually an interval or union of intervals). Among all the sets of possible values, we must choose one that we think represents the most extreme evidence against the hypothesis. That is called the critical region of the test statistic. The probability of the test statistic falling in the critical region when the hypothesis is correct is called the alpha value of the test. After the data is available, the test statistic is calculated and we determine whether it is inside the critical region. If the test statistic is inside the critical region, then our conclusion is either the hypothesis is incorrect, or an event of probability less than or equal to alpha has occurred. If the test statistic is outside the critical region, the conclusion is that there is not enough evidence to reject the hypothesis.

The significance level of a test is the maximum probability of accidentally rejecting a true null hypothesis (a decision known as a Type I error).For example, one may choose a significance level of, say, 5%, and calculate a critical value of a statistic (such as the mean) so that the probability of it exceeding that value, given the truth of the null hypothesis, would be 5%. If the actual, calculated statistic value exceeds the critical value, then it is significant "at the 5% level".

5 Types of hypothesis tests

• Parametric tests of a single sample:
  • t test
  • z test
• Parametric tests of two independent samples:
  • two-group t test
  • z test
• Parametric tests of paired samples:
  • paired t test
• Nominal/ordinal level test of a single sample:
  • chi-square
  • Kolmogorov-Smirnov one sample test
  • runs test
  • binomial test
• Nominal/ordinal level test of two independent samples:
  • chi-square
  • Mann-Whitney U
  • Median
  • Kolmogorov-Smirnov two sample test
• Nominal/ordinal level test for paired samples:
  • Wilcoxon test
  • McNemar test

6 Reliability and validity

Research should be tested for reliability, generalizability, and validity. Generalizability is the ability to make inferences from a sample to the population.

Reliability is the extent to which a measure will produce consistent results. Test-retest reliability checks how similar the results are if the research is repeated under similar circumstances. Stability over repeated measures is assessed with the Pearson coefficient. Alternative forms reliability checks how similar the results are if the research is repeated using different forms. Internal consistency reliability checks how well the individual measures included in the research are converted into a composite measure. Internal consistency may be assessed by correlating performance on two halves of a test (split-half reliability). The value of the Pearson product-moment correlation coefficient is adjusted with the Spearman-Brown prediction formula to correspond to the correlation between two full-length tests. A commonly used measure is Cronbach's α, which is equivalent to the mean of all possible split-half coefficients. Reliability may be improved by increasing the sample size.

Validity asks whether the research measured what it intended to. Content validation (also called face validity) checks how well the content of the research are related to the variables to be studied. Are the research questions representative of the variables being researched. It is a demonstration that the items of a test are drawn from the domain being measured. Criterion validation checks how meaningful the research criteria are relative to other possible criteria. When the criterion is collected later the goal is to establish predictive validity. Construct validation checks what underlying construct is being measured. There are three variants of construct validity. They are convergent validity (how well the research relates to other measures of the same construct), discriminant validity (how poorly the research relates to measures of opposing constructs), and nomological validity (how well the research relates to other variables as required by theory) .

Internal validation, used primarily in experimental research designs, checks the relation between the dependent and independent variables. Did the experimental manipulation of the independent variable actually cause the observed results? External validation checks whether the experimental results can be generalized.

Validity implies reliability : a valid measure must be reliable. But reliability does not necessarily imply validity :a reliable measure need not be valid.

7 Types of errors

Random sampling errors:

• sample too small
• sample not representative
• inappropriate sampling method used
• random errors

Research design errors:

• bias introduced
• measurement error
• data analysis error
• sampling frame error
• population definition error
• scaling error
• question construction error

Interviewer errors:

• recording errors
• cheating errors
• questioning errors
• respondent selection error

Respondent errors:

• non-response error
• inability error
• falsification error

Hypothesis errors:

• type I error (also called alpha error)
  • the study results lead to the rejection of the null hypothesis even though it is actually true
• type II error (also called beta error)
  • the study results lead to the acceptance (non-rejection) of the null hypothesis even though it is actually false

8 See also

• marketing research

9 List of related topics

• list of marketing topics
• list of management topics
• list of economics topics
• list of finance topics
• list of accounting topics

Marketing Psychometrics

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