## Importance of Research

## Asking a Research Question

## Introduction

## IRB

## Research Strategies

## Variables in Research

## Research Design

## Research Designs Examples

## Methods Section

## Statistics Crash Course

**Basic Concepts**

- Scales
- Nominal Data
- categorized using text values (i.e., discrete), with no implied order among the categories
- male first followed by female
- nominal data are existential—it either exists or fails to exist
- those who have previously worn hearing aids and another for those who have never worn devices
- A respondent who selects the second check box indicates no experience with amplification, nullifying the first option (i.e., experience with hearing aids).

- Ordinal Data
- indicates a greater than/less than relationship, but the data does not provide information about the relative difference between each data category
- Letter grades (A, B, C, D, and F) assigned to students for their academic performance is an example of ordinal data
- The grade of A is better than the grade of B, B is better than a C, and so on
- Now, assume that Lisa earned an A and Jill earned a B in the same course. The data indicates that Lisa performed better than Jill, but it is not known by how much (i.e., was it by 6% or 2% points?).
- Likert scale used to assess a listener's loudness growth
- rates perceptual loudness by providing a number associated with a descriptor (e.g., 1 = too soft, 2 = soft, 3 = comfortable, 4 = loud, 5 = too loud)
- The data are nominal because there are greater than/less than distinctions, with no evidence revealing by how much the distinctions differ. Findings from a loudness growth task imply that increases in amplitude cause a change in perceptual loudness for that listener. Therefore, ordinal data are coded using text values, or discrete values.

- Interval Data
- improvement over the nominal and ordinal scales, where values are ordered as a constant (i.e., continuous) scale, allowing for the comparison of differences between points
- For example, the Fahrenheit scale of temperature has a freezing point of 32° and a boiling point of 212°. The difference is 180°, which is spaced in equal intervals. Therefore, the difference between 20° and 40° F is equal to the difference between 75° and 95° F.

For the interval scale, however, the zero point is arbitrary - the starting and ending points of the scale are not absolute values, but rather, relative values
- Because the values used are relative, the responses described by the data indicate only the magnitude of the relationship between the variables. For instance, 60° F is hotter than 30° F. It is inappropriate to say, however, that 60° F is twice as hot as 30° F.

Two examples of instruments in Audiology created with an interval scale are the APHAB and the speech intelligibility rating (SIR) test (Cox & McDaniel, 1989). The APHAB is a 24-item self-assessment inventory in which patients report the amount of problems they are having with communication or aversiveness to loud sounds in various everyday listening situations. Patients who complete the APHAB rate their everyday listening experiences using a Likert scale, with each descriptor having a percentage (A = always [99%], G = never [1%]).

For the SIR test, subjects listen to a short passage of connected speech presented against noise through a hearing aid. While listening to the speech signal, the subject's task is to generate a perceptual rating of intelligibility using an interval scale ranging between 0 and 10. The 0 value on the scale indicates that connected speech was completely unintelligible and the 10 indicates that connected speech was fully intelligible. The scale assumes that each value, between 0 and 10, is equally spaced, with no true zero.

- Ratio Data
- The ratio scale has all the properties of the nominal, ordinal, and interval scales. In addition, the zero point of this scale has a fixed starting point. As a result, the difference between the intervals is always measured from a zero, or reference, point. As an example, consider the concept of time. The difference between 3:00 and 5:00 is the same as the difference between 8:00 and 10:00. With respect to ratio comparison, 10 hours is twice as long as 5 hours.

The decibel (dB) is an example of a unit measured on the ratio scale. The decibel increases in equal intervals in a logarithmic manner and the ratio is always compared with its reference point (i.e., 10−12 watts/m2, 20 μPa). With respect to ratio, an increase of 3 dB results in twice the power (in dB IL) and an increase of 6 dB results in twice the pressure (in dB SPL).

- The ratio scale has all the properties of the nominal, ordinal, and interval scales. In addition, the zero point of this scale has a fixed starting point. As a result, the difference between the intervals is always measured from a zero, or reference, point. As an example, consider the concept of time. The difference between 3:00 and 5:00 is the same as the difference between 8:00 and 10:00. With respect to ratio comparison, 10 hours is twice as long as 5 hours.

- Nominal Data
- Variables
- any entity that can take the form of different numeric or text values. independent vs dependent is relevant when investigating cause-effect relationships.
- age is a numeric value
- age becomes a text value when numeric values are categorized by group (child, adult, older adult)
- Independent
- the treatment, or what is being manipulated
- must have at least two different treatment methods
- ex: microphones: omni-directional microphone and directional microphone
- ex: loudspeaker array: loudspeaker positioned at 180 and 8 speakers separated by 45
- ex: better ear: open and plug & muff

- Dependent
- the outcome of interest, what is being measured, which changes in response to independent variable
- ex: dB outcome, benefit in percent outcome

- Discrete Variables
- text values are limited to being integers (whole numbers) only
- ex: child = 1, adult = 2, older adults = 3
- would be incorrect if 0.1, 0.2, or 0.3 had been used

- Continuous
- can take any numerical - including decimal places - within a defined range

- Independent

- any entity that can take the form of different numeric or text values. independent vs dependent is relevant when investigating cause-effect relationships.

- Random Sampling
- it is ideal to select randomly a large enough sample from a larger population
- helps to eliminate bias

**Measures of Central Tendency**

- Mean
- average, sum of scores divided by the number of scores

- Median
- score corresponding to 50% of the observations when arranged in order
- n+1/2

- Mode
- the most commonly occurring score

**Measures of Variability**

- Dispersion (Variability)
- the degree to which individual data points are distributed around the mean

- Range
- the distance from the lowest to the highest

- Interquartile range
- the range of the middle 50% of the observations

- Trimmed samples
- samples with a percentage of extreme scores removed

- Trimmed statistics
- statistics calculated on trimmed samples

- Variance
- sample variance
- population variance

- The Standard Deviation
- the positive square root of the variance and for a sample
- it is the square root of the variance
- it is calculated by the square root of the sum of the total minus the average (X - Xavg) squared divided by the sample size minus 1 (n-1)

- The Mean and the Variance as Estimators
- we generally calculate measures such as mean and the variance as estimates of the corresponding values in populations
- sample variance offers an excellent example of property estimators known as bias
- bias - a property of a statistic whole long-range average is not equal to the parameter it estimates

- Boxplots: a Graphical representation of dispersion and extreme scores
- boxplot - a graphical representation of the dispersion of a sample
- quartile location - median location + 1/2
- whisker - line from top and bottom of the box to the farthest point that is no more than 1.5 times the H-spread from the box

The Normal Distribution

- a specific distribution having a characteristic bell-shaped curve
- abcissa - horizontal axis
- ordinate - vertical axis
- density - heigh of the curve for a given value of X
- closely related to the probability of an observation in an interval around X

- standard normal distribution
- a normal distribution with a mean (average) equal to 0 and variance = to 1 denotes as N(0,1)
- transformed normal distribution is calculated as Z = X - u / standard deviation
- Z-score - number of standard deviations above or below the mean

- measures related to Z
- z formula can be used to convert a distribution with any mean and variance to a distribution with a mean of 0 and standard deviation (and variance) of 1
- referred to as
**standard scores** - percentile: the point below which a specified percentage of the observations fall

Basic Concepts of Probability

Sampling Distribution and Hypothesis Testing

Regression

Multiple Regression

Hypotesis Tests Applied to Means

Power

Analysis of Variance (ANOVA)

Choosing appropriate statistical tools

Chi Square (Time Permits)

Non-Parametric and Distribution Free Tests (Time permits)

- an event can occur in X ways and can fail in Y ways; 5 clear marbles 5 dark marbles, event of picking a clear marble
- probability that it will occur is X/X+Y or 5/10
- probability that it will fail is Y/X+Y or 5/10
- analytic view - definition of probability in terms of analysis of probable outcomes
- sample with replacement - sampling in which the item drawn on trial N is replaced before the drawing on trial N + 1
- event - outcome of the trial; trial 1: clear marble trial 2: dark marble

- independent events - events are independent when the occurrence of one has no effect on the probability of the other
- mutually exclusive - when the occurrence of one makes the other even impossible from happening
- joint probability - probability of the co-occurrence of two or more events
- conditional probability - the probability of one event given the occurrence of some other event
- unconditional probability - the probability of one event ignoring the occurrence or nonoccurrence of some other event
- examples: two dice in a bag one black one white
- prob of 1 black = 1/2
- prob of 1 white = 1/2
- prob of one 6 = 2/12 = 1/6
- prob of one 6 and one 1 at the same time = 1/6 X 1/6
- prob of two 6s at the same time = 1/6 X 1/6
- prob of one 6 and it's black = 1/2 X 1/6
- prob of one 6 and it's black and 1 and it's white at the same time = 1/6 X 1/2 X 1/6 X 1/2

- discrete vs continuous variables - distributions of two kinds of variables are treated somewhat differently in probability theory
- discrete variables - we speak of the probability of the outcome
- continuous variables - we speak of the probability of obtaining a value that falls within a specific interval

Sampling Distribution and Hypothesis Testing

- sampling error - variability of a statistic from sample to sample due to chance
- hypothesis testing - a process by which decisions are made concerning the values of parameters
- sampling distribution - the distribution of a statistic over repeated sampling from a specified population
- standard error- the standard deviation of a sampling deviation
- sampling distribution of the mean - the distribution of sample means over repeated sampling from on population
- research hypothesis - the hypothesis that the experiment was designed to investigate
- null hypothesis - the statistical hypothesis tested by the statistical procedure; usually a hypothesis of no difference or no relationship
- sample statistics - statistics calculated from a sample and used primarily to describe the sample
- descriptive statistics

- test statistics - the results of a statistical test
- inferential statistics

- Using the Normal Distributions to Test Hypothesis
- decision making - a procedure for making logical decisions on the basis of sample data
- rejection level - the probability with which we are willing to reject the null hypothesis when it is in fact correct
- significance level - the probability with which we are willing to reject the null hypothesis when it is in fact correct
- rejection region - the set of outcomes of an experiment that will lead to rejection of the null hypothesis
- alternative hypothesis

- Type I and Type II Errors
- power: the probability of correctly rejecting a false null hypothesis
- critical value - the value of a test statistic at or beyond which we will reject the null hypothesis
- Type I error - the error of rejecting the null hypothesis when it is true alpha the probability of a Type I error
- Type II error - the error of not rejecting the null hypothesis when it is false - the probability of a type II error

- power: the probability of correctly rejecting a false null hypothesis
- One and Two Tailed Tests
- one tailed test - a test that rejects extreme outcomes in one specified tail of the distribution
- directional test - another name for a one-tailed test

- two tailed test - a test that rejects extreme outcomes in either tail of the distribution
- non-directional test - a test that rejects extreme outcomes in either tail of the distribution

- one tailed test - a test that rejects extreme outcomes in one specified tail of the distribution

- Correlation
- Correlation - relationship between variables
- Correlation coefficient - a measure of the relationship between variables
- Pearson product-moment correlation coefficient - the most common correlation coefficient

- The Covariance
- covariance = a statistic representing the degree to which two variables vary together
- Cov (x,y) = the sum (X - Xmean)(Y - Ymean) / n-1

Regression

Multiple Regression

Hypotesis Tests Applied to Means

- One Sample
- Two Related Samples
- Two Independent Samples

Power

Analysis of Variance (ANOVA)

- One-Way ANOVA
- Repeated Measures ANOVA

Choosing appropriate statistical tools

Chi Square (Time Permits)

Non-Parametric and Distribution Free Tests (Time permits)