Statistical analysis Prepared and gathered by Alireza Yousefy(Ph.D)

What is Meant by Statistics? Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions

Types of Statistics Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way.

Types of Statistics Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

Scales of measurement Nominal Ordinal Interval Ratio

Nominal scale Numbers represent labels, identify categories Not really a scale at all Example: number codes for religious affiliation: 1=protestant, 2=catholic, 3=Islamic etc.

Levels of Measurement Nominal scale Nominal measurement consists of assigning items to groups or categories. No quantitative information is conveyed and no ordering of the items is implied. Nominal scales are therefore qualitative rather than quantitative. Examples: Religious preference, race, and gender are all examples of nominal scales Statistics: Sum, Frequency Distributions

Ordinal scale On a continuum Numbers only tell you the order in which observations fall -- ranks Know nothing about the size of the interval between numbers Examples: class rank, rank on a “best movies” scale

Measurements with ordinal scales are ordered: higher numbers represent higher values. However, the intervals between the numbers are not necessarily equal. There is no "true" zero point for ordinal scales since the zero point is chosen arbitrarily. For example, on a five-point Likert scale, the difference between 2 and 3 may not represent the same difference as the difference between 4 and 5. Also, lowest point was arbitrarily chosen to be 1. It could just as well have been 0 or -5. Ordinal Scale

Interval scales Continuous scale in which equal intervals between values represent equivalent “amounts” However, ratios of values are not valid and there is no true zero point Many scales in psychology are treated as interval scales Examples of interval scales: IQ number, Exam number …

Ratio scales All the properties of interval scales In addition, ratios make sense, e.g., 2 Kg is twice the weight of 1Kg True zero point, e.g., zero weight, zero money Tend to be concrete, tangible things, e.g., number of events, money, weight

Interval & Ratio Scales On interval measurement scales, one unit on the scale represents the same magnitude on the trait or characteristic being measure across the whole range of the scale. For example, on an interval/ratio scale of anxiety, a difference between 10 and 11 would represent the same difference in anxiety as between 50 and 51.

Assumption: 90% of analyses will use following procedures 1) Cross tab with  2 2) t-test -- actually optional, since ANOVA can accomplish same things, but useful to know about 3) Analysis of Variance (ANOVA) 4) Simple Correlation/Regression 5) Multiple Regression

Some additional analyses could be useful: we could learn them depending on time  2 Goodness of Fit Tests Spearman correlation Factor analysis/principal component analysis Nonparametric analogues of common parametric tests

Steps in planning analysis 1) Examine your data –Exploratory data analysis 2) Choose analysis based on characteristics of the data and your research questions -- see following slides

Questions to answer to plan your analysis 1) Type of data: Categorical or measurement? 2) Type of research question: Focus on differences or relationship 3) # of Groups or Variables 4) Independence (if relevant): Are your measurements independent or dependent?

Type of data Qualitative categorical Quantitative measurement Type of question Type of question Are frequency of each levels of One categorical variable fit With expected frequency? Are two categorical Variables relevant? Goodness of fit  2 Contingency table  2 Relationship Difference Number of prediction Number of groups 2 More than 2 1 2or more Multiple regression measurement continuous rank Spearman R Pearson correlation I-Dep Dep IndependentT Mann Whitney U Paired T willcoxon I-DEP DEP Number of Ind variables 1 2 or more 1 way Anova Kruskal-Wallis Repeated Measures Friedman Factorial Anova

A STATISTICALLY SIGNIFICANT DIFFERENCE MEANS large -- NO, not necessarily. Even a small mean difference between two groups could be statistically significant if the sample size is large enough. 2. of practical significance -- NO. Statistical significance has to do with mathematical probability. It has nothing to do with whether a research result is meaningful or useful. 3. not likely to be due to sampling error -- YES. What you are saying is that the difference is larger than the expected amount of sampling error. 4. likely to be due to the treatment -- YES. There are two main reasons for differences between "equivalent groups" -- one is sampling error and the other is the treatment that was received by one group but not the other. In this case, the difference is larger than what we would expect from sampling error.

A STATISTICALLY SIGNIFICANT DIFFERENCE MEANS-2 5. generalizable -- YES. Probability says that the difference should be replicable in another sample taken from the same population. 6. true -- If true means correct, absolute, 100%, then NO. We cannot be absolutely certain that a difference exists in the population - ever. A statistically significant difference simply means that probability is on our side, based on evidence taken from a sample. 7. too large to have occurred by chance -- YES. "Chance" refers to sampling error. We can estimate the level of chance and see (after the treatment) if the difference is even larger than that. If it is, then we attribute the difference to the effects of the treatment.