t-test

• 1. test


  An inferential statistical test used to determine if there is a significant difference between the means of two groups

  
• To understand this chapter, you need to have understood

  • The mean, standard deviation and standard error
  • Z-scores and the normal distribution
  • Assumptions underlying the use of parametric tests
  • Probability distributions like the t-distribution
  • One- and two-tailed hypotheses
  • Statistical significance
  • Confidence interval

  
• Topic objectives


  • Describe and explain the nature of t-test, including its definition, importance, and assumptions
  • Differentiate between an independent and paired t-tests
  • Interpret outputs for t-test
  • Analyze published works which used t-test
• Topic outline


  • 7.1 The nature and importance of t-test
  • 7.2 Hypothesis Testing with t statistic
  • 7.3 Measuring effect size for the t statistic
  • 7.4 Confidence Intervals
  • 7.5 Assumptions of t-test
  • 7.6 Comparing Repeated- and Independent-Measures Designs
  • 7.7 Independent t-test
  • 7.8 Paired t-test
  • 7.9 Non-Parametric Alternatives

  
• Example scenarios


  • Is The Conjuring 1 scarier than The Conjuring 2? We measure heart rates (indicating anxiety) during both films and compare them
  • Is psychotherapy method 1 better than psychotherapy method 2?
  • Does background noise affect short term memory? Noise? Or no noise?
  • Is work better when you listen to your favorite music? Or when you work in silence? Compare the grades
• We can expose different entities to different experimental manipulations (a between-groups or independent design), or take a single group of entities and expose them to different experimental manipulations at different points in time (a repeated measures or within-subject design)
• Independent samples t-tests


  Used when you want to compare the mean scores of two different groups of people or conditions

  
• Paired samples t-test


  Used when you want to compare the means scores for the same group of people on two different occasions or when you have matched pairs

  
• Types of t-tests


  • Independent-measures t-test
  • Between-subjects t-test
  • Unpaired t-test
  • Student's t-test
  • Dependent samples t-test
  • Paired-difference t-test
  • Matched pairs t-test
  • Repeated-samples t-test

  
• Analysis of two conditions includes

  • Descriptive Statistics
  • Inferential tests
  • Effect Size
  • Confidence limits

  
• Rationale for the t-test

  1. 2 samples of data are collected and the sample means calculated

  2. If the samples come from the same population, then we expect their means to be roughly equal

  3. We compare the difference between the sample means that we collected to the difference between the sample means that we would expect to obtain if there were no effect

  4. If the difference between the samples we have collected is larger than we would expect based on the standard error then one of two things has happened

  
• Possible outcomes if the difference between the samples is larger than expected


  • There is no effect but sample means from our population fluctuate a lot and we happen to have collected 2 samples with very different means
  • The 2 samples come from different populations, which is why they have different means
  • There is genuine difference between samples

  
• Study designs


  • Determining if there are differences between 2 independent groups
  • Determining of there are differences between interventions
  • Determining if there are differences in change scores
• Effect divided by error

  The basis for the t-test statistic

  
• Changes and differences


  • We are often interested in change over time or changes in response to some manipulation
  • We measure a single variable at different times
  • We are interested in the change, or the difference score
• Matched pairs


  We test to see if people who are matched or paired in some way agree on a specific topic
• Null hypothesis for related samples

  There is no change or difference

  
• Alternative hypothesis for related samples


  • There is a change or difference
  • The average score increases
  • The average score decreases

  
• Assumptions of paired t-test

  • Level of Measurement: Interval or ratio level (continuous)
  • Normally distributed
  • The difference between the two scores obtained for each subject should be normally distributed

  
• SPSS


  Analyze → Compare means → Paired Samples T-test

  
• Calculating the effect size

  • You can calculate the Cohen's d
  • You can calculate the eta squared
• Interpretation of effect size

  • Cohen's d: 0.2 = small, 0.5 = medium, 0.8 = large
  • Eta squared: 0.01 = small, 0.06 = medium, 0.14 = large
• Fear of Statistics Test (FOST)


  Test that measures students' scores on their fear of statistics
• There was a statistically significant decrease in FOST scores from Time 1 (M = 40.17, SD = 5.16) to Time 2 (M = 37.5, SD = 5.15), t (29) = 5.39, p <.001 (two-tailed)
• The mean decrease in FOST scores was 2.67 with a 95% confidence interval ranging from 1.66 to 3.68
• The eta squared statistic (.50) indicated a large effect size

  
• In a normal population you would expect the verbal IQ (VIQ) and the performance IQ (PIQ) of people with chronic illness to be similar. The population mean IQ is 100
• Batool and Kausar (2015) hypothesised that there are likely to be significant changes in the pre- and post-diagnostic health-related behaviours of patients with hepatitis C. The sample was 100 patients diagnosed with hepatitis C. A questionnaire was used to assess these behaviours, in particular the researchers assessed medication adherence of the patients
• Null Hypothesis (Ho)


  The population means of the two groups are equal (μ1 = μ2) OR There is no significant difference between the 2 groups (μ1 = μ2)
• Alternative Hypothesis (H1)


  The population means of the two groups are not equal (μ1 ≠ μ2) OR There is a significant difference between the 2 groups (μ1 ≠ μ2)
• Pooled variance


  Formula used to calculate the variance of two groups
• Twenty-four people were involved in an experiment to determine whether background noise (music, slamming of doors, people making coffee, etc.) affects short-term memory (recall of words). Half of the sample were randomly allocated to the NOISE condition, and half to the NO NOISE condition. The participants in the NOISE condition tried to memorize a list of 20 words in two minutes, while listening to pre-recorded noise through earphones. The other participants wore earphones but heard no noise as they attempted to memorize the words

  
• Confident limits around the mean

  Exploratory data analysis to look at descriptive statistics
• Confidence Intervals


  Statistical measure used to estimate the range in which the true population parameter is likely to fall
• Measure of Effect


  Calculation to find the effect size (d) using the formula: d = (x1 - x2) / mean SD
• Size of Effect


  Interpretation of the effect size:

  0.2 = small

  0.5 = medium

  0.8 = large
• Independent Samples T-Test


  Statistical test used to determine if there is a statistically significant difference in the mean scores for two independent groups

  
• Assumptions for Independent Samples T-Test:

  1. One Dependent Variable (continuous)

  2. One independent variable (2 categorical, independent groups)

  3. Independence of Observations

  4. No outliers in any of your independent groups

  5. Approximately Normally distributed

  6. Homogeneity of Variances

  
• Independence of Observations


  There is no relationship between the observations in each group of the IV or between the group themselves
• Homoscedasticity (Homogeneity of Variances)


  Assumption of equal or similar variances in different groups being compared, assessed through Levene's test