PSYC 523- Statistics

Question 1

ANOVA

Answer 1

Analysis of variance: a statistical technique used to compare whether three or more populations are statistically different from each other; determines whether there is a significant difference between the groups but does not reveal where that difference lies - must do further tests to determine.

Clinical example: A group of psychiatric patients are trying three different therapies: counseling, medication, and biofeedback. You want to see if one therapy is more efficacious than the others. You will gather data and run an ANOVA on the three groups- counseling, medication, and biofeedback- to see if there is a significant difference between any of them.

Question 2

Clinical v. statistical significance

Answer 2

Clinical significance refers to the meaningfulness of change in a client’s life due to the treatment. Do the patient’s symptoms reduce in a meaningful or noticeable way? Does the quality of life improve for the patient?

Statistical significance refers to whether or not a treatment made a statistically significant impact on some outcome variable of interest or the impact of the treatment had a high probability of not being due to chance alone. A treatment can be statistically significant in research but not clinically significant.

Clinical example: If a randomized controlled trial does not show that a treatment is more effective than no treatment or a placebo, but that treatment produces a meaningful difference in a client’s life, it could be said to have clinical but not statistical significance.

You are trying to decide between two treatments for your client with treatment-resistant depression. One has demonstrated high clinical significance and high statistical significance in RCTs. The other shows high statistical significance but low clinical significance. You choose the one with high clinical significance because it assesses treatment efficacy from the patient perspective.

Question 3

Construct validity

Answer 3

In research design, construct validity is the degree to which a test or study measures the qualities or the constructs that it is claiming to measure.

There are two ways of collecting evidence for construct validity, both of which are statistical procedures: convergent validity is how well a certain measure of a construct correlates with other well-established measures of that construct and divergent validity is how well the measure of a construct does not correlate with measures of other constructs.
In order to have high construct validity, a test should correlate highly with measures of the same construct (convergent validity) and not correlate highly with measures of other constructs (divergent validity).

Clinical example: If people score significantly differently on a new test designed to measure intelligence compared to a recognized test of intelligence, the new test may be lacking construct validity.

A group of researchers create a new test to measure depression. They want to ensure that the test has construct validity, in that it actually measures the construct of depression. To do this, they measure how much the test correlates with the Beck Depression Inventory and how much it does not measure another concept like anxiety.

Question 4

Content validity

Answer 4

In research design, content validity is the degree to which a measure or study includes all of the facets/aspects of the construct that it is attempting to measure. Content validity cannot be measured empirically but is rather assessed through logical analysis.

Clinical example: A depression scale may lack content validity if it only assesses the affective dimension of depression (emotion related- decrease in happiness, apathy, hopelessness) but fails to take into account the behavioral dimension (sleeping more or less, eating more or less, energy changes, etc) Because of this therapists, end up using other scales.

Question 5

Correlation v. causation

Answer 5

In the context of research, correlation means that a relationship exists between two variables. This relationship can be positive or negative; coefficient will fall between -1.00 and +1.00. Causation means that a change in one variable affects a change in the other variable. Causality is usually determined via controlled studies, when you can isolate variables you want to examine and control for extraneous variables. Correlation does not indicate causation.

Clinical example: A study found that minutes spent exercising correlated with lower depression levels. This study was able to show that depression levels and exercise were correlated, but could not go so far as to claim that one causes the other.

Question 6

Correlational research

Answer 6

Research method that examines the potential for relationships between variables that might logically seem to be related. The technique identifies a mathematical relationship and does not establish causal factors.

• Produces correlation coefficient; ranges from 1.0 to -1.0 depending on strength/direction of the relationship between the two variables
• Very common in psychological research; usually cost-effective
• PROS - inexpensive, produces wealth of data, encourages future research; precursor to experiment determining causation
• CONS - cannot establish causation or control for confounds
• Statistical tests include Pearson, Spearman, & point-biserial

Clinical example: Shelia’s patient Donna suffers from an anxiety disorder. She brings Shelia an article claiming that eating out of plastic containers causes cancer. After reading the article, Shelia explains that the study referenced in the article is a correlational study, which only shows that there is a relationship between eating out of plastic containers and cancer, but it does not prove that eating out of plastic containers causes cancer.

Question 7

Cross-sectional design

Answer 7

A type of research that simultaneously compares individuals of different ages at one specific point in time. This type of design is very common and used in online surveys.

• Groups can be compared across a variety of dependent variables
• Advantages include a collection of large amounts of data in a short amount of time & low cost
• Drawbacks included the inability to infer causation (because it is just a snapshot)
• Considered quasi-experimental design (participants are not selected randomly - selected based on age)

EXAMPLE: George was looking to study the difference in peer relations and self-esteem in various age groups. He decided to use a cross-sectional design comparing 6 year-olds, 12-year-olds, 18-year-olds, and 25-year-olds.

EXAMPLE: You’re treating someone with depression. He is having a hard time finding the energy to carry out daily activities. The therapist shows him a cross-sectional study looking at depression levels and the utilization of behavioral activation. Specifically, the effectiveness of taking daily walks to increase energy level. The therapist explains that those who walk daily have been shown to have lower depression and higher energy levels, especially for his age group.

Question 8

Dependent t-test

Answer 8

In psychological research, a type of statistical analysis that compares the means of two groups where the values in one sample affect the values in the other sample. Because the sample is carried across the test (AKA matched pairs or repeated measures), they are dependent on one another.

• Used when the design involves matched pairs or repeated measures, and only two conditions of the independent variable
• It is called “dependent” because the subjects carry across the manipulation–they take with them personal characteristics that impact the measurement at both points—thus measurements are “dependent” on those characteristics.

Clinical example: A researcher wants to determine the effects of caffeine on memory. They administer a memory test to a group of subjects have the subjects consume caffeine then administer another memory test. Because they used the same subjects, this is a repeated-measures experiment that requires a dependent t-test during statistical analysis.

Question 9

Descriptive v. inferential

Answer 9

Descriptive statistics are those which are used to describe and summarize a data set.

• Can only be used to describe the sample they are conducted on.
• Common tools include measures of central tendency, variance, and skew.
• We choose a group that we want to describe and then measure all subjects in that group

Inferential statistics take data from a sample and make inferences about the larger population from which the sample was drawn.

• Need to have confidence that sample accurately reflects the population (population must be defined) → importance of random sampling
• Common techniques include hypothesis testing, regression analysis, etc.
• The statistical results incorporate the uncertainty that is inherent in using a sample to understand an entire population.

EXAMPLE: A researcher conducts a study examining the rates of test anxiety in Ivy League
students. This is a descriptive study because it is concerned with a specific population. However, this study cannot be generalized to represent all college students, so it is not an inferential
study.

Question 10

Double-blind study

Answer 10

A type of experimental design in which both the participants and the researchers are unaware of who is in the experimental condition and who is in the placebo condition.

• In contrast to a single-blind study, where only the participants are unaware of who is in the experimental condition.
• Double-blind studies eliminate the possibility that the researcher may somehow communicate (knowingly or unknowingly) to a participant which condition they are in, thereby contaminating the results.

Example: A study testing the efficacy of a new SSRI for anxiety is using a double-blind study. Neither the experimenter nor the participants are aware of who is in the treatment group and who is receiving a placebo. This setup ensures that the experimenters do not make subtle gestures accidentally signaling who is receiving the drug and who is not, and that experimenter expectations could not affect the studies outcome.

Question 11

Ecological validity

Answer 11

The extent to which an experimental situation approximates the real-life situation which is being studied.

• Researchers call for these in hopes they will better generalize to the real-world
• Different from external validity
• Experiments high in ecological validity tend to be low in reliability because there is less control of the variables in real-world like settings

EXAMPLE: A researcher wants to study the effects of alcohol on sociability, so he administers beer to a group of subjects and has them interact with each other. To increase their ecological validity, he decides to carry out the study in an actual bar.

Question 12

Effect size

Answer 12

Part of: research methods and statistical analysis

What: A quantitative measure of the strength of a relationship between two variables; refers to magnitude of an effect.

• It is also valuable for quantifying the effectiveness of a particular intervention, relative to some comparison - commonly used in meta-analyses
• Effect size can be used with the correlation between two variables, regression coefficients or the mean difference.

Example: A researcher conducts a correlational research study on the relationship between caffeine and anxiety ratings. The study produces a correlation coefficient of 0.8 which is considered a large effect size. The effect size reflects a strong relationship between the caffeine and anxiety.

Question 13

Experimental research

Answer 13

A form of research in which one variable (the independent variable) is manipulated in order to see what effect it will have on another variable (the dependent variable). Researchers try to control any other variables (confounds) that may affect the dependent variable(s). Experimental research is the only way to establish causation.

Example: A researcher conducts an experimental research study to examine the relationship between caffeine intake and anxiety ratings. The study administers various levels of caffeine (the independent variable) to the low, high, and no caffeine groups. The participants are then asked to report their anxiety levels (the dependent variable). They found that those who had more caffeine reported feeling more anxious.

Question 14

Hypothesis

Answer 14

In the field of research, a hypothesis is a formally stated prediction that can be tested for its accuracy.

• Essential to the scientific method and testing in research
• Hypotheses help to focus the research and bring it to a meaningful conclusion.
• Without hypotheses, it is impossible to test theories.
• Specifically, a hypothesis is a statement or proposition about the characteristics or appearance of variables, or the relationship between variables, that acts as a working template for a particular research study.

EXAMPLE: A famous hypothesis in social psychology was generated from a news story, when a woman in New York City was murdered in full view of dozens of onlookers. Psychologists John Darley and Bibb Latané developed a hypothesis about the relationship between helping behavior and the number of bystanders present, and that hypothesis was subsequently supported by research. It is now known as the bystander effect.

Question 15

Independent t-test

Answer 15

Statistical analysis that compares the means of two independent groups, typically taken from the same population (although they could be taken from separate populations).

• Determines if there is a statistical difference between the two groups’ means
• We make the assumption that if randomly selected from the same population, the groups will mimic each other; the null hypothesis is no difference between the two groups

EXAMPLE: Fred is analyzing the best treatment options for his patient Harold. He reads a study comparing two different types of therapies. After utilizing an independent t-test, the researchers found that there was not a statistically significant difference between the treatment options. Harold decides that both are good options for his patient and he decides to think about his client’s person variables that might make one better than the other.

Question 16

Internal consistency

Answer 16

Part of: psychological research

What: the extent to which different items on a test measure the same ability or trait; intercorrelations among items on the same test

-usually measured with Cronbach’s alpha but can use split-half or KR20

EXAMPLE: Patient comes in with symptoms of PTSD. You decide to search for a psychological test that is designed to help you to detect and diagnose PTSD. You come across the Posttraumatic Stress Diagnostic Scale (PDS). The test manual indicates that the PDS is a valid measure of PTSD. You look in the test manual of the PDS and find that Cronbach’s alpha is 0.91. This indicates that the PDS has strong internal consistency.

Question 17

Internal validity

Answer 17

The extent to which the observed relationship between variables in a study reflects the actual relationship between the variables. Internal validity is a measure of the integrity of a study. Internal validity is how sure we can be that the experimental treatment was the only cause of change in a dependent variable(s). Control for confounding variables can increase internal validity, as well as a random selection of participants.

EXAMPLE: Researchers investigated a new tx for depressing using tight controls in terms of who could be a participant. For instance, they did not allow anyone with comorbidity to participate. This increased the study’s internal validity. It did, however, jeopardize the ecological validity of the research.

Question 18

Interrater reliability

Answer 18

In research design, interrater reliability is a type of reliability that measures the agreement level between independent raters. It is useful with measures that are less objective and more subjective. This type of reliability is used to account for human error in the form of distractibility, misinterpretation or simply differences in opinion.

EXAMPLE: Three graduate students are performing a natural observation study for a class that examines violent video games and behavior in a group of 9 year old boys. The students rated the behavior on a scale of 1 (not aggressive) to 5 (very aggressive). However, the responses were not consistent between the observers. The study lacked inter-rater reliability.

Question 19

Measures of central tendency

Answer 19

Provide statistical descriptions of the center of the distribution; describes a data set.

• Three main measures are used: the mean, mode and median.
• These help to summarize the main features of a data set and identify the score around which most scores fall.
• The mean is the arithmetic average of all scores within a data set.
• The mode is the most frequently occurring score.
• The median is the point that separates the distribution into two equal halves.
• The median and mode are the most resilient to outliers.

EXAMPLE: A researcher is studying the frequency of binge eating in a group of girls suffering from binge eating disorder. To better understand the data that was gathered, they start by calculating the measures of central tendency: the most frequently occurring number of episodes in the group, the average number of episodes, and the number of episodes in the middle of the set. In other words, the mode median and mean.

Question 20

Measures of variability

Answer 20

In statistics, measures of variability are measures of how scores in a distribution vary around the central tendency. Three primary measures: range, variance and standard deviation. The range is obtained from taking the two most extreme scores and subtracting the lowest from the highest. The variance is the average squared deviation around the mean, and must be squared because the sum of the variations would equal zero. The standard deviation is the square root of the variance and is highly useful in describing variability.

EXAMPLE: A researcher is studying the frequency of binge eating in a group of girls suffering from binge eating disorder. After calculating the measure of central tendency, they decide that they want to know more about the distribution of number of episodes. They decide to calculate the measures of variability. This includes the range, variance, and standard deviation

Question 21

Nominal/ordinal/interval/ratio measurements

Answer 21

These are 4 types of measurements seen in statistics.

-Nominal scales are scales in which labels are assigned for identification but cannot be counted or categorical data where there may be more than two categories. They have none of the 3 properties that distinguish scales (no magnitude, equal intervals or abs 0) .Ex: male/female; Republican, Democrat
-Ordinal scales have data (numbers) that indicate order only, but may not indicate what measurement was used to determine the order or the magnitude of the differences within the order. Ordinal scales are used for rankings of individuals or variables and have the property of magnitude.
-Interval scales have true score data where you know the score a person made and you can tell the actual distance between individuals based on their respective scores, but the measure used to generate the score has no true zero. Thus, they have magnitude and equal intervals between any two observations, but do not have the property of absolute zero.
Ex: most psychological measures, IQ, SAT, GRE
-Ratio scales have interval data with an absolute zero. They have all three properties of scales including magnitude, equal intervals and absolute zero.
The type of scales used dictates what statistical procedures may be run on a data set.

EXAMPLE: A researcher is creating a questionnaire to measure depression. They include nominal
scale questions (“what is your gender?”) ordinal scale questions (“rank your mood today from 1-very unhappy to 5-very happy”) and ratio scale questions (“how many hours of sleep do you get on average?”)

Question 22

Normal curve

Answer 22

A normal curve is a normal distribution, graphically represented by a bell-shaped curve.

• A frequency distribution where most occurrences take place in the middle of the distribution and taper off on either side
• All measures of central tendency are at the highest point of the curve
• Based on infinity
• Symmetrical, extremes are at the tails, further from the center=lower frequency, divisible into deviations, fits any set of data where n=infinity

EXAMPLE: A researcher is developing a new intelligence test. After obtaining the results, they found that the scores fell along a normal curve: most participants scored in the middle range with very few obtaining either the highest or lowest scores (scores were normally distributed).

Question 23

Probability

Answer 23

A mathematical statement indicating the likelihood that something will happen or that a particular event will occur when a particular population is randomly sampled, symbolized by (p). The higher the p value, the more likely that the phenomenon or event happened by chance. Probability is based on hard data (unlike chance); p is between 0 and 1

EXAMPLE: Researchers are conducting a study on the heritability of bipolar disorder. They find
that there is a strong genetic link, meaning there is a greater probability of an individual having the disorder if one of their parents also has it.

Question 24

Parametric v. nonparametric statistical analyses

Answer 24

Part of psychological research
What: Parametric statistical analyses are inferential procedures that require certain assumptions about the distribution of scores. They are usually used with scores most appropriately described by the mean, they are based on symmetrical distributions or distributions that come close to symmetry, they focus on 1 variable or relationship, and are robust procedures with negligible amounts of error.

Nonparametric statistical analyses involve inferential procedures that do not require stringent assumptions about the parameters of the raw score population represented by the sample data and are usually used with scores most appropriately described by the median or the mode. Nonparametric data have skewed distributions.

Parametric analyses are preferred because they have greater statistical power and are more likely to detect statistical significance.

EXAMPLE: Researchers set up a study to determine if there is a correlation between hours of sleep per night and ratings of happiness. Because they used a very small sample, they cannot assume the data are symmetrically distributed and therefore must use a nonparametric test.

Question 25

Quasi-experimental research

Answer 25

Part of: research methodology

What: Experiments that lack random assignment. Research designs that study quasi-experimental variables or variables where random assignment is not feasible or ethical. For eample variables that could cause potential harm i.e. depression, smoking; subjects cannot be randomly assigned to them due to health/ethical concerns.

Why: Quasi-experimental studies cannot fully control for loss of internal or external validity.

EXAMPLE: Researchers want to conduct a study examining how opioid addiction affects depression. Because they cannot ethically assign the condition of opioid addiction to their participants, they must place the participants in groups according to whether they are already addicted or not. Opioid addiction is a quasi-experimental variables, qualifying this study as quasi experimental research.

Question 26

Random sampling

Answer 26

The process of selecting a sample randomly from the population to better represent the entire group as a whole; that is, all members of the population being studied have an equal chance of being chosen/sampled

• Two random samples from same population should have similar means; the mean of a random sample is a good estimate of population mean
• Helps control for confounding variables

EXAMPLE: A researcher is doing an experiment on college students and must select a sample of students from a larger population. To ensure that they are not biased in their selection of students, they assign each student a number and then randomly draw numbers to create the sample. This is an example of random sampling.

Question 27

Regression

Answer 27

Part of: research design

What: A inferential statistical technique in which one looks at the scores on one variable or set of variables to predict the score of a dependent variable of interest

Essentially, an estimate of what percentage of the variance in the dependent variable can be accounted for by the independent variable(s)

Expressed as the line of best fit, which minimizes the sum of the distances of the data points from the line (y’ = a + bx)

Can also help identify the most important contributors/factors for the prediction

EXAMPLE: A developmental psychologist performed a study on aggressive behavior in boys and hormone levels. Researchers performed a regression analysis on the data. Their results showed that the severity and frequency of the boys’ aggression could be accurately predicted based on the levels of testosterone.

Question 28

Sample v. population

Answer 28

A sample is a relatively small subset of the population that is selected to represent the population in a study; important that the sample be representative of the population being studied.

A population is all members of a group; the larger group of individuals from which a sample is selected.

EXAMPLE: Researchers want to conduct a study to investigating how opioid addiction affects depression. As it would be nearly impossible to study every single individual with an opioid addiction, they select a sample of individuals that closely represents the whole population. To ensure that the sample is representative, they compare the sample and population means.

Question 29

Scientific methodology

Answer 29

Part of: research design

Who: Basic guidelines formalized by Francis Bacon that support all legitimate research.

What:

Starts with, primarily, three tenets/assumptions,

1. nature is lawful - not random
2. laws of nature can be identified and understood through research
3. behavior is viewed as deterministic - influenced by identifiable internal and external events

4 Steps
conceptualize process/problem to be studied → collect relevant data through research → analyze the data → draw conclusions based on analyses

Data should be:

• empirical (measurable),
• objective,
• systematically gathered,
• controlled,
• and verifiable by others (study duplication);

EXAMPLE: A researcher wants to understand the relationship between caffeine and sociability. First, they form a hypothesis that caffeine consumption increases sociability. Next, they conduct an experiment and collect the relevant data. Then they analyze their results. Finally, they draw the conclusion that caffeine increases sociability, based on their results.

Question 30

Standard error of estimate

Answer 30

In regression analysis, this is a measure of accuracy of predictions made:

• How much the data points are spread around the regression line
• Also referred to as standard error of the residuals - a residual is the difference between the observed value of the dependent variable (Y) and the predicted value (Y’)
• Standard deviation of residuals is the standard error of estimate

EXAMPLE: A researcher wants to find out if there is a relationship between social media usage and depression. They collect data and find that there is a positive relationship between the two variables. Next, they calculate the regression line. Next, they want to know how accurate the predictions made using the regression equation are, so they calculate the standard error of estimate.

Question 31

Standard error of measurement

Answer 31

The standard deviation of errors of measurement that is associated with the test scores for a specified group of test takers.

• In simpler terms- an estimate of how much an individual’s score would be expected to change if retesting with same/equivalent form of test. (The smaller the SEM the more precise the measurement capacity of the instrument)
• Common tool in psychological research and standardized testing
• Reminds the evaluator that there is error and that the test score is just an estimate not exact
• Has an inverse relationship with the reliability coefficient
• Used to calculate confidence intervals or a range around the estimate “true” score

EXAMPLE: A researcher develops a test to measure depression, then administers it to a sample. They want to analyze the data that they gathered using statistics. They calculate the SEM and it turns out to be low which indicates that the measurement is fairly precise. They then decide to carry out further statistical analysis.

Question 32

Standard error of the difference (2 sample t-test)

Answer 32

Part of: statistical analysis

What: the estimated standard deviation of the differences between the means of two independent samples, meaning it’s the estimate of error between the two groups. A two-sample t-test compares the means of two samples to see if they came from the same population.

(diff - subtract 2 things, compare means 2 samples- did they come from the same population?)

Example: A researcher conducts a study on how caffeine affects test scores. They take the mean of scores from each group (with or without caffeine) and calculate the differences between the means. They then used the standard error of the difference to find the amount of error between the estimated and actual difference.

Question 33

Standard error of the mean (single sample z-test)

Answer 33

Part of: statistical analysis

What: compares a random sample back to the population if the population mean and standard deviation are known (as opposed to a single sample t-test, wherein we don’t know the population standard deviation).

The standard error of the mean is the average of the deviations of sample means around the population mean.

Comparison can be done before experimental manipulation to make sure the sample is representative, or it can be done after manipulation

EXAMPLE: A researcher creates a new test to measure intelligence, which they test on a random sample. Because they know the mean IQ and the SD of the population, they run a single sample z test by calculating the standard error of mean. This confirms that their sample is indeed representative of the population.

Question 34

Standard error of the mean, estimated (single sample t-test)

Answer 34

Part of: research and statistical analysis

What: The average of sample means around a population mean. Standard error of mean, estimated, is the basis of a single sample t-test.

Single sample t tests compare a single random sample back to the population, wherein the population mean is known, but the population standard deviation is unknown, so it uses sample standard deviation to estimate population standard deviation. (In contrast to a single sample z-test, which is used when both population mean and population standard deviation are known.)
Comparison can be done before experimental manipulation to make sure the sample is representative, or it can be done after manipulation.

Example: A researcher creates a new eating disorder measure, which they test on a random sample. Because they know the population mean but not the population standard deviation, they use the sample standard deviation to make an estimate. They run a single sample t-test by calculating the standard error of mean, estimated, which compares the random sample back to the population.

Question 35

Standardization sample

Answer 35

Related to test development- a large sample of test takers who represent the population for which the test is intended; also called norm group
-Results used to establish normal distribution & norms
○ Use in comparison for future test scores; norms are not standards of performance, but serve as a frame of reference for test score interpretation
-Can be an issue if standardization samples do not include culturally diverse clients
-In general, increasing standardization sample size adds to psychometric soundness

EXAMPLE: You’re a child psychologist that administers IQ tests to all of your patients. You compare all of the scores with those of a standardization sample in order to determine whether a pt’s score is above or below average. Issues can arise for culturally diverse clients, though, as they were not always included in the standardization sample.

Question 36

Statistical significance

Answer 36

Related to statistical analysis and hypothesis testing; the likelihood that results did not occur by chance.
-Criterion of significance is chosen, typically 0.05; means that researcher is 95% confident the obtained results are real and not due to randomness/error - called significance level
-P-value (represents probability) is found and compared to significance level; if p < or equal to significance level, indicates significance
■ Reject null hypothesis

EXAMPLE: A new drug for anxiety is being tested. The researchers have set the p-value for the research to be p

Question 37

Type I and Type II error

Answer 37

Two types of errors seen in research. Random sampling and increased sample size help avoid these errors.

Type I error occurs when researcher rejects a null hypothesis when it is actually true; detecting an effect that is not present

• researchers incorrectly conclude that the independent variable(s) had an effect on the dependent variable(s)
• Aka “false positive”
• How to minimize/avoid Type I error: Increase significance level to higher threshold i.e. from 0.05 to 0.01

Type II error occurs when a researcher does not reject a null hypothesis when it is not true; failure to observe a difference when in truth there is one.

• researchers incorrectly conclude that the independent variable(s) had no effect on the dependent variable(s)
• Aka “false negative”
• How to minimize/avoid Type II error: Increase sample size, Increase significance level to higher threshold i.e. from 0.05 to 0.01

EXAMPLE: A researcher is testing a new drug for PTSD. After reviewing the results, they concluded that the drug effectively reduced symptoms; however, the conclusion was wrong and the drug had no impact. This is an example of Type I error because the researchers rejected the null hypothesis of no difference between the tx and control groups when in fact there was no difference .