Review of Basic Statistics and Terminology
- 1. Review of Basic Statistics and Terminology What are Statistics? There are various steps to consider in scientific research. The key steps are: • Research Design • Data Collection • Description of data and statistical analysis (our focus for this course) (Daniels, D., Nizam, Z, 1999, Biostatistics (notes), RSPH) All of these steps consider directly or indirectly data or the statistics used to do scientific research. There are actually many different definitions of statistics but the one that is more general and fits most situations is the following: Statistics are produced from data. The dictionary definition of "statistics" refers to “numeric indicators of nations. Popular usage of the term points to numeric summaries that condense information, or numbers that are used to make comparisons, or numbers that portray relationships or associations. The term statistics also refers a formal discipline of study. The field of statistics is the science of generalization. Built upon theories of probability and inference, statistics support the making of broad generalizations from a smaller number of specific observations.” There are two types of statistics researchers are usually most interested in: Descriptive and Inferential. Descriptive Statistics “give us information around discovering and describing the important features and trends contained in a set of data using quick and easy graphical and numerical methods. In other words, they are most often used to describe populations. Even though descriptive statistics most often yield information about one variable, the relationship between two variables can be described as well. With inferential statistics, research hypotheses about a population of interest are investigated using information on the basis of measurements contained in a random sample of data from the population.“ (Price, I., 2000, University of New England, School of Psychology) An important concept to understanding descriptive statistics is the Shape of Distribution. A variable’s distribution shape is a very important aspect of the variables description. The shape of the distribution tells you the frequency of values from different ranges of the variable. Four common terms to understand when evaluating shape of the distribution is: • Normal • Skewness • Kurtosis • Symetrical/Asymmetrical
- 2. A normal distribution is a bell shaped curve that is symmetrical with scores more frequent in the middle. A standard normal distribution has a mean of 0 and a standard deviation of 1. Skewed distributions yield most scores as being high or low. Usually, a small percentage of scores are shown in one direction away from the majority. Skewed distributions produce what is known as a “tail.” If the “tail” points toward the upper end of the score continuum, the distribution is “positive.” If the “tail “points toward the lower end of the score continuum, the tail is “negative.” (Huck, S.W. & Cormier, W.H., 1996. Reading Statistics and Research, New York, New York, p. 31). Distributions that are skewed are typically considered asymmetrical while normal distributions are symmetrical. Kurtosis measures “peakedness” of the distribution. A high peak gives a distribution with “fat” tails and a low even distribution while a low peak gives a “skinny” tail and a distribution concentrated towards the mean. (http://www.statsoft.com/textbook/stbasic.html, Readings: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm http://davidmlane.com/hyperstat/normal_distribution.html Common concepts/terms A Population in statistics is often referred to as an entire group of people or objects. When a statistical inference is made, a selection, called a sample, is first removed from a larger group called a population. The sample, usually comprised of people or objects, is then measured using statistics. These measurements are then summarized. A guess is then made as to the numerical value of the same statistical concept in the population. (Huck, S.W. & Cormier, W.H., 1996. Reading Statistics and Research, New York, New York, p. 100). Correlations – A correlation is a measure of the relation between two or more variables. The most popular technique for assessing the strength of a bivariate (having two variables) relationship is Pearson's product-moment correlation, also called linear or product-moment correlation. Correlation coefficients can range from -1.00 to +1.00. The value of -1.00 represents a negative correlation while a value of +1.00 represents a perfect positive correlation. A value of 0.00 represents a lack of correlation. In order to evaluate a correlation between 2 variables, it’s important to know the “magnitude” as well as the significance of the correlation. A perfect correlation rarely exists but a correlation at .6 or above is often considered to be satisfactory. Sometimes correlations observed at .3 are significant and worth reporting. Also, a negative correlation does not infer an unsatisfactory correlation, as a -.129 can be a significant correlation. A negative correlation really means there is an indirect or inverse relationship between the variables. Readings: http://www.statsoft.com/textbook/stbasic.html (Basic Statistics tab - Correlation section) http://www.uwsp.edu/psych/stat/7/correlat.htm (Lessons I, II, III, and V)
- 3. Outliers – Outliers in a set of data, have a value so far removed from other values in the distribution that its presence cannot be attributed to the random combination of chance causes. An outlier is an atypical value that does not belong to the distribution of the rest of the values in the data set. Values that are more than 2.5 standard deviations from the mean can be defined as outliers. This would hold when the distribution is unimodal and symmetrical. For skewed data, median is a better indicator of central location than mean. In such a case, an observation can be considered as an outlier if it more than 1.5 inter-quartile range away from the closest quartile. An outlier would be labeled an extreme outlier if it is more than 3 inter-quartile range from the closest quartile, and it would be called mild otherwise. Frequencies - A frequency distribution yields how many subjects or objects were similar and ended up in the same category or had the same score. The total frequency is usually labeled as N, and tells us how many subjects were measured. T-Tests – the t-test is the most commonly used method to evaluate the differences in means between two groups. For example, the t-test can be used to test for the difference in knowledge scores between a group of students who participated in a sex education class and those students who did not. As long as variables are normally distributed within each group and variation of scores in the two groups is not reliably different, small sample sizes can be used when performing a t-test. The p-level reported with a t-test represents the probability of error involved in accepting our research hypothesis about the existence of a difference. Reading: http://www.texasoft.com/tutorial-statistics-compare-2-groups.htm Confidence Intervals – A Confidence Interval is a range of values that is normally used to describe the uncertainty around a point estimate of a quantity, for example, a mortality rate. Confidence intervals provide a means of assessing and reporting the precision of a point estimate and account for the uncertainty that arises from the natural variation inherent in data collection. Confidence intervals are constructed by adding a specific amount to the statistic (upper limit) and by subtracting a specific amount from the statistic (lower limit). Researchers also attach a percentage to any interval constructed, usually either 95 or 99. Reading: http://www.stat.yale.edu/Courses/1997-98/101/confint.htm Working with Data and questions It’s important to be prepared for working with data before you enter, manipulate, or analyze data. This also involves knowing which program you will use for data management and analysis. Part of the preparation involves knowing your questions. What questions do you want answered from the data? What are your key research questions? Data
- 4. collection should be systematic and done with purpose, even though it’s sometimes easy to explore data first, especially if you are working with an existing data set or a large data set. If you are creating a data collection instrument, it should be designed using specific techniques. When designing the instrument, it’s important to think about how you are going to analyze each question and what exactly you want to gain from each question. Several factors to think about when creating a survey is the following: Variable name – Unique identifiers for each question. Often times, variable names should not be longer than 8 characters. This is because some statistical programs only allow 8 characters and may pose a problem if importing/exporting between programs if more than 8 characters exist for variable names. Also, variables names usually don’t begin with numbers or contain symbols, punctuation, or spaces. Lastly, variables names should relate to the question and be easy to identify or recall. Level of measurement – Refers to nominal, ordinal, or interval/ratio level of measurement. Each variable will have one of these levels of measurement. This is important to know because the level of measurement often depends on how we analyze our data. We will discuss this more in the independent/dependent variable lab week. Type of variable – Field types refer to the data types associated with a variable in your survey and vary from program to program but the two most common field types are numeric, or integer and string or text variables. Numeric variables often contain whole numbers. String variables contains letters, numbers, or special symbols. In survey development, we may wish to use text responses for certain questions but when entering data, associate the text responses with numbers and therefore would be entered and analyzed as a numeric variable. For example, if we have a question with a Likert scale such as: 1. I enjoy exercise. Strongly Agree Agree Neutral Disagree Strongly Disagree Then we may want to associate these responses with numbers such as: 1 - Strongly Agree 2 - Agree 3 - Neutral 4 - Disagree 5 - Strongly Disagree
- 5. Types of Data – Terms associated with types of data are Quantitative and Qualitative. Quantitative data refers to data that are numeric. Qualitative data refers to data that are non-numeric. The way we analyze these types of data are very different. Quantitative data can be further categorizedinto discrete and continuous data. Discrete data are numeric data that has a finite number of possible values. Continuous data is numeric data that have infinite possibilities. Qualitative data are also referred to as categorical data, meaning non-numerical data which may be divided into groups. We will also discuss this more in the independent/dependent variable lab week. Labels – Refers to a description of the variable or the response categories. There are variable labels and value labels. Variable labels assigns descriptive labels to all variables. Variable names and labels may be the same for some variables. Often, a label might be the entire question or a part of the question. Value labels give a description of the response categories. The value label assigns descriptive labels in the data file. Not all variables necessarily have assigned value labels. Codebook – A codebook displays a information for the variables in a data set and is a description of the data that was collected. According to the Data and Statistical Services at Princeton University, the best codebooks have: Description of the study: who did it, why they did it, how they did it. 1. Sampling information: what was the population studied, how was the sample drawn, what was the response rate. 2. Technical information about the files themselves: number of observations, record length, number of records per observation, etc. 3. Structure of the data within the file: hierarchical, multiple cards, etc. 4. Details about the data: columns in which specific variables can be found, whether they are character or numeric, and if numeric, what format. 5. Text of the questions and responses: some even have how many people responded a particular way. Review of Research Designs Every research project should begin with a purpose. When thinking about the purpose of the research, you will also want to think about the research design, or the structure of the study. Research designs fall into 2 categories:
- 6. Experimental – a study or experiment that imposes a treatment on a group of subjects or objects in order to observe the response. This differs from observational studies, involving collecting and analyzing data about a group of subjects or objects, without imposing treatment or changing existing conditions. A “true” experimental design has the following characteristics: 1) random assignment of participants to groups, and 2) manipulation of an internal variable Quasi-experimental – Controls for confounding variables, used to investigate cause-and-effect relationships. This is different than experimental because there is no random assignment of participants to groups. Readings: http://www.socialresearchmethods.net/kb/quasiexp.php Study hypotheses Once a problem or area of interest has been identified and researched, a hypothesis is then created and stated by the investigator. Hypothesis testing is a “statement that describes a phenomenon and the relationships between the variables in the problem” (Southeastern Institute for Training and Evaluation). In order to formulate a hypothesis test, usually some theory has been researched and stated. The hypothesis is used for several reasons: 1. Gives a statement of relationships between variables to be tested 2. Offers a possible explanation of the research problem and a focus for the testing 3. Provides direction for the research 4. Provides a framework or organization for reporting the findings of the study (Southeastern Institute for Training and Evaluation) Hypothesis Testing There typically is a 6 step process involved in hypothesis testing: • State the null hypothesis – a statement as to the unknown quantitative value of the parameter of interest. The null hypothesis relates to the statement being tested. • State the alternative hypothesis – the predicted relationship between the variables of interest. The alternative hypothesis relates to the statement to be accepted if/when the null is rejected. • Select a level of significance – Fixed probability of wrongly rejecting the null hypothesis, if it is in fact true. Usually the significance level is chosen to be .05 or 5%...often seen as p<.05. • Collect and summarize the sample data – process used to move from the beginning points of the hypothesis testing procedure to the final decision • Refer to a criterion for evaluating the sample evidence – involves asking the question, are the sample data inconsistent with what would likely occur if the
- 7. null hypothesis is true? If yes, the null hypothesis would be rejected. • Make a Decision to discard/retain the null hypothesis (Huck, S.W. & Cormier, W.H., 1996. Reading Statistics and Research, New York, New York, p. 150). More information on hypothesis testing can be found in the readings. Readings: http://davidmlane.com/hyperstat/logic_hypothesis.html http://www.stats.gla.ac.uk/steps/glossary/hypothesis_testing.html Analyzing Data There are many programs, procedures and commands used to analyze data. For the purposes of this class, only a select group of data analysis procedures will be used to analyze data using SPSS software. In order to analyze data, it must first be collected and entered into a database. Most analysis programs will allow you to import a variety of database files. Setting up databases is not within the scope of this class and will not be discussed. I will show you exporting and importing using various file formats at the closing on-campus session. Data cleaning – If you enter data and set up your own database, it’s important to make sure that the data is clean and examined for possible data errors. Common errors may be: • Values entered outside of the response categories • Missing data • Unexpected responses that are not easily identified • Spelling errors (Southeastern Institute for Training and Evaluation) Reading: http://www.tulane.edu/~panda2/Analysis2/datclean/dataclean.htm Basic Data Analysis This section will mainly discuss basic descriptive statistics. Measures of Central Tendency Mean - The mean is a descriptive statistic probably used most often as a measure of central tendency. Means are often reported in conjunction with confidence intervals. Confidence intervals for the mean yield a range of values around the mean where the “true” expected mean is found. For example, if the mean in your sample is 60, and the lower and upper limits for a P=.05 confidence interval are 53 and 67, then you can conclude that there is a 95% probability that the population mean is greater than 53 and lower than 67. (http://www.mste.uiuc.edu/hill/dstat/dstat.html) Median – The median is the middle value of a distribution when the data are arranged from highest to lowest. Exactly half the data is above the median and half is below it. If there is an even number of values, then it becomes the average of the two middle observations.
- 8. Mode – The most frequently occurring value. It is possible to have more than one mode or no mode at all. Reading: https://statistics.laerd.com/statistical-guides/measures-central- tendency-mean-mode-median.php Measures of Variability around the mean After reviewing a set of values, measures of central tendency and shape of the distribution, it’s important to know more about the variability of a set of values. Most groups of values have some degree of variability, meaning at least some of the values differ (vary) from one another. There are two main measures of variability. Variance – The variance is the sum of squared deviations from the mean divided by N-1. The variance is simply the standard deviation squared. Standard deviation – Determined by figuring out how much each score deviates from the mean and by placing these deviation scores into a formula. The standard deviation is the square root of the variance. Readings: https://statistics.laerd.com/statistical-guides/measures-of-spread- range-quartiles.php Cross Tabulations Cross tabulations show the relationship between 2 variables. In most programs, cross tabulations are used to get basic descriptive statistics to conduct other tests, such as a chi-square. Different programs may use different commands and/or procedures to conduct a cross tabulation. When performing a cross tabulation, it’s important to know your independent and dependent variables since independent variables are used as column headings and dependent variables are found in the rows Reading: http://web.idrc.ca/en/ev-56452-201-1-DO_TOPIC.html