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RD&Statistics > Week 5: Comparing Means: Independent and Paired T-test > Flashcards

Week 5: Comparing Means: Independent and Paired T-test Flashcards

1
Q

What are the 3 types of t-tests? - (3)

A
  1. One-samples t-test
  2. Paired t-test
  3. Independent t-test
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2
Q

What is the decision framework for choosing a paired-sample (dependent) t-test? - (5)

A
  1. What sort of measurement = continous
  2. How many predictor variables = one
  3. What type of predictor variables = categorical
  4. How many levels of categorical predictor = two
  5. Same or different participants for each predictor level = same
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3
Q

What is the decision framework for choosing independent-t-test? (5)

A
  1. What sort of measurement = continous
  2. How many predictor variables = one
  3. What type of predictor variables = categorical
  4. How many levels of categorical predictor = two
  5. Same or different participants for each predictor level = different
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4
Q

Whats a one-sample t-test?

A

Compares the mean of the sample data to a known value

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5
Q

What is the assumptions of one-sample t-test? - (4)

A
  • DV = Continous (interval or ratio)
  • Independent scores (no relation between scores on test variable)
  • Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk
  • Homogenity of variances
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6
Q

Example of one-sample t-test RQ - (2)

A

Is the average IQ of Psychology students higher than that of the general population (100)?

A particular factory’s machines are supposed to fill bottles with 150 millilitres of product. A plant manager wants to test a random sample of bottles to ensure that the machines are not under- or over-filling the bottles.

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7
Q

What is the assumptions of independent samples t-tests (listing all of them) - (7)

A
  1. Independence. – no relationship between the groups
  2. Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk
  3. Equal variances
  4. Homogeneity of variances (i.e., variances approximately equal across groups) via non significant Levene’s test
  5. DV = Interval or continuous
  6. IV = Categorical
  7. No significant outliers
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8
Q

What is an RQ example of independent samples t-tesT?

A

Do dog owners in the country spend more time walking their
dogs than dog owners in the city?

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9
Q

What is the assumptions of paired t-test? (listing all) - 3

A

DV is continuous

Related samples: The subjects in each sample, or group, are the same. This means that the subjects in the first group are also in the second group

Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk

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10
Q

What is an example of RQ of paired t-test?

A

Do cats learn more tricks when given food or praise as positive feedback?

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11
Q

If we are comparing differences between means of two groups in independent/paired t-test then all we are doing is

A

predicting an outcome based on membership of two groups

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12
Q

Indepdnent and paired t-tests can fit into an ideal of a

A

linear model

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13
Q

What is coding with dummy variables and an example?

A

E.g., coding absence of cloak in terms of numbers (like 0) and pps with clock as 1 even though it is a categorical variable

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14
Q

Independent and paired t-tests (comparing difference of two means) fit into idea of general linear model

What does b0 and b1 represent in this general linear model? - (2)

A

*b0 is equal to the mean of group coded as 0 (in this case no cloak)
* b1 is difference between group means –> difference between cloak and no cloak

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15
Q

The t-distributed is defined by its

A

degrees of freedom - related to the sample size.

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16
Q

The t distribution has heavier tails for - (2)

A

lower degrees of freedom (small N studies)

increased uncertainty and a higher likelihood of observing extreme values than large N studies with less heavy tails as t distribution goes to normal

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17
Q

The probability of a value of t occurring yields the p value for the difference between the means occurring by

A

chance

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18
Q

What are the two different t-tests? - (2)

A
  • Independent-means t-test
  • Dependent/Paired -means test
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19
Q

Independent and Paired T-tests have one predictor (X) variable with 2 levels and only …. outcome variable (Y)

A

one

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20
Q

When is an independent-means t-test used?

A

When 2 experimental conditions and different participants are assigned to each conditiont

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21
Q

What is independent-means t-test sometimes called as well?

A

independent-samples t-test

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22
Q

When is a dependent-means t-test used?

A

Used when there are 2 experimental conditions and same participants took part in both conditions of the experiment

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23
Q

What is dependent-means t-test sometimes referred to?

A

Matched pairs or paired samples t-test

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24
Q

For independent and paired t-tests we compare between the sample means that we collected to the difference between sample means that we would expect if

A

there was no effect (i.e., null hypothesis was true)

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25
Q

In independent and paired t-tests if standard error was small then suggests that sample means of two groups are quite

A

similar

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26
Q

Formula of calculating t- test statistic (form depend on whether same or different participants used in each experimental condition) in independent/paired

A
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27
Q

Formula of calculating t-statistic shows obtaining t-test statistic by diving the model/effect by the in independent /apried

A

error in the model

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28
Q

Expected difference in calculating t-test statistic in most cases is

A

0 - expect differences between sample group means we colelcted to be different to 0

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29
Q

If observed difference between sample means get larger in t-tests then more confident we become that

A

null hypothesis is rejected and two sample means differ because of experimental manipulation

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30
Q

Both independent t-test and paired t-test are … tests based on normal distribution

A

parametric tests

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31
Q

Since independent and paired t-tests are parametric tests they assume that the - (2)

A
  • Sampling distribution is normally distributed - in paired it means sampling distribution of differences of scores is normal not the socres itself!
  • Data measured at least interval level
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32
Q

Since independent-tests used to test different groups of people it also assumes - (2)

A
  • Variances in populations are roughly equal (homegenity of variance) = Leven’s test
  • Scores are independent since they come from different people
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33
Q

Diagram of equation of calculating t-statistic from paired t-test and explain - (2)

A
  • Compares mean differences betwen our samples (–D) to the differences we would expect to find between population means (uD) which is divided by standard error of differences (sD / square root N)
  • If H0 is ture, then expect no difference between population means hence uD = 0
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34
Q

A small standard error of differences tells us that in paired-t-test

A

pairs of samples from a population have similar means to population

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35
Q

A large standard error of differences tells us that in paired t-test - (2)

A

that sample means can deviate quite a lot from the populatio mean and

sampling distribution of differences is more spread out

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36
Q

The average difference between person’s socre in condition 1 and condition 2 -(¯D) in paired t-test is an indicator of

A

systematic variation in the data (represents experimental effect)

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37
Q

If average differences (–D) between our samples is large and standard error of differences is small in paired-t test then we can be confident that

A

the difference we observed in our sample is not a chance result and caused by experimental manipulation

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38
Q

How do we normally calculate the standard error?

A

SD divided by square root of sample size

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39
Q

How to calcuate the standard error of differences in paired-test?(σ –D)

A

Standard deviation of differences divided by square root of sample size

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40
Q

the t-statistic in paired t-test is

A

ratio of systematic variation in experiment (average difference D) and unsystematic variation (standard erro of differences)

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41
Q

When would we expect t statistic greater than 1 in paired-t-test equation?

A

If the experimental manipulation creates any kind of effect,

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42
Q

When would we expect t statistic less than 1 in paired t-test equation?

A

If the experimental manipulation is unsuccessful then we might expect the variation caused by individual differences to be much greater than that caused by the
experiment

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43
Q

In pairered and generally independent t-tests we can compare the obtainee value of t against thmaximum value we would expect to get by chance alone in t distribution with same DFs and if value we obtain exceeds the

A

critical value then conflict if reflects an effect in our IV

44
Q

What does this output mean?

A

Sleep condition scored mean of 65.38 and no sleep had mean of 60.22

45
Q

In Paired Samples Correlation box here we would expect that

A

someone’s score for first condition would be associated in second condition

46
Q

What does this paired samples correlation show?

A

people doing well in first exam likely doing well in second exam regardless of condition they are in and significantly correlated (r= 0.664)

47
Q

What does this SPSS output show? = paired t- test

A

t(19) = 2.72, p = 0.012

48
Q

What does negative t-value mean?

paired t-test.

A

First condition had smaller mean than second condition

49
Q

What does 95% confiderence interval of difference mean in SPSS output of paired t-test?- (3)

A
  • 95% of the samples (e.g., if we had 100 samples then 95 of those samples..) the constructucted CIs contain true value (population) of the mean difference
  • CIs tell us boundaries within which true mean difference is likely to lie
  • The true value of mean difference is unlikely to be 0 if Cis does not contain 0
50
Q

How to calculate effect size for independent and paired t-tests?

A

Using cohen’s D

51
Q

Diagram of calculating Cohen’s D Statistic for sleep vs no sleep for paired

A

Minus big mean from small mean divided by smallest SD (control group)

52
Q

What is interpretation of Cohen’s D? - (3)

A
  • Around 0.20 = a small effect
  • Around 0.50 = a medium effect
  • Around 0.80 & above = a large effect
53
Q

What does Cohen’s d of 0.20 represent

A

difference between groups is a 1/5 of SD

54
Q

Diagram of writing up paired t-test result

A
55
Q

To calculate effect size for independent and paired t-tests, beside Cohen’s D, we can also

A

calculate effect size r (above 0.50 is large effect) by converting t-value to r-value

56
Q

With independent t-test there are two different equations that can be used depending on whether the samples

A

contain an equal number of people

57
Q

With independent t-test since different participants participate in different condition, the pairs of scores will differ not just of experimental manipulation but also because of

A

other sources of variance (such as individual differences between participants’ motivation, IQ etc..)

58
Q

With dependent t-test we look at differences between pairs of scores because

A

scores came from same participant and so individual differences were eliminated

59
Q

Equation of independent t-test of equal N sizes for each condition

A
60
Q

Equation of independent t-test of equal N sizes becomes like the final form since - (3)

A
  • We are looking at differences between the overall means of 2 samples and compare with differences we would expect to get between means of 2 populations from which sampels come from
  • If H0 is true, samples drawn from same population
  • Therefore under H0, u1 = u2 therefore u1 - u2 = 0
61
Q

Equation of independent t-test in numbers for equal N sizes

A
62
Q

We use variance of sum law to obtain the estimate of standard error for each … in independent t-test equation for equal N sizes

A

sample group

63
Q

What does variance sum of law state?

A

variance of the sampling distribution is equal to the sum of the variances of the two populations from which the samples
were taken

64
Q

This independent t-test standard error formula combines the

A

standard error for two samples

65
Q

In independent t-test when we want to compare two groups that contain different number of participants then equation … is not appropriate

A
66
Q

For comparing two groups with unequal number of participants in independent t-test then we use the

A

pooled variance estimate t-test

67
Q

The pooled variance estimate t-test is used which takes into account of the

A

differnece in sample size by weighting the variance of each sample

68
Q

Formula of pooled variance estimate t-test - (2)

A

Each variance of sample is multipled by its DF and added together and divided by the sum of weights (sum of two DFs)

Larger samples better than small ones as close to population

69
Q

In formula of pooled variance estimate t-test it weights the variance of each sample by the

A

number of degrees of freedom (N-1)

70
Q

As with dependent t-test we compare obtained value of t in independent sample against the

A

maximum value we would expect to get by chance alone in t distribution with same DFs

71
Q

As with the dependent t-test we can compare the obtained value of t against the maximum
value we would expect to get by chance alone in a t-distribution with the same degrees of freedom

if value we obtain exceeds this critical value then

A

we can be confident that this reflects an effect of our independent variable

72
Q

What does this output show? - in independent t-test - (2)

A

Sleep condition scored an average exam score of 66.200 and no sleep condition earned an average of 58.73

Effect size (Cohen’s D) = Mean of sleep minus mean of no sleep divided by standard deviation of sleep (control grp) = 66.20-58/73/7.12

73
Q

In independent samples t-test we check for Levene’s test for quality of variances which determine whether

A

we got equal variance across the groups or whether the variances are unequal

74
Q

In independent t-test, Levene’s test we are looking for a non-significant p-value which shows that

A

no statistically significant difference in variances between the two groups - report results from equal variances assumed

75
Q

In independent t-test if Levene’s test was significant then it means that

A

variances between the 2 groups are different and they are statistically significantly different - report data from equal variances not assumed

76
Q

What does this output show in independent t-test? - (2)

A
  • Levene’s test is not significant (p = 0.970) so no stats sig differences in variance between two groups
  • t(28) = 2.87, p = 0.008
77
Q

Diagram of reporting independent t-test

A
78
Q

Paired vs independent t-tests - who has better power?

A

Paired t-t ests

79
Q

Since paired-t-tests use same participants across conditions the … is reduced dramatically compared to independent t-test

A

unsystematic variance

80
Q

The non-parametric counterpart of dependent t-test is called

A

Wilcoxon signed rank test

81
Q

The non-parametric tests of the independent t-test is

A

Wilcoxon rank sum test and Mank Whitney test

82
Q

What does this SPSS output of independent-test of Levene’s show

A

homogeneity of variance as assessed by Levene’s Test for Equality of Variances (F = 1.58, p = .219)

83
Q

What does this independent samples t-test output show? (DV = puppes avergae weight gain between 12 and 28 weeks of age and IV= diet A, B)

A

This study found that puppies in diet B had statistically significantly higher average daily weight gain (89.29 ± 9.93 g/day) between 12 and 28 weeks of age compared to puppies in diet A (60.20 ± 6.85 g/day), t(27)= -9.24, p < .001.

84
Q

In Cohen’s D 3 types of SD can be used in the formula … but… - (4)

A
  • Pooled SD (over conditions)
    Averaged SD
    Control group SD
    But they make very small difference
85
Q

Cohen’s d for diet was 4.25

Is this a:
Small effect
Medium effect
Large effect

A

Large effect

86
Q

There is two ways of computing effect size for independent and paired

A

r and cohen’s D

87
Q

Although there are 2 ways to compute effect, use

A

Cohen D

88
Q

In independent t-test, we no longer have

A

have difference values that are paired, we have only two sample means from independent samples.

89
Q

With independent test the challenge is to

A

standardize the difference in means to compute a t statistic.

90
Q

With independent-test formula, we - (2)

A

The general idea still holds - we divide by an estimate of the standard error

The formula for the standard error here combines the errors for the two samples, but it only holds for samples of equal size

91
Q

An independent t-test compares

A

Compares the differences between means of groups containing different participants when the sampling distribution is normal, the groups have equal variances and data are at least interval

92
Q

What does paired t-test compare the means of

A

two related groups on the same continuous, dependent variable.

93
Q

Another example of two samples independent t-test scenario

RQ

Sample

DV

Hypothesis

Test

Sig

  • (6)
A

Research question: Which of the two diet formulas is better for puppies?

Sample: 15 were randomly assigned to each of the two diets (A and B).

Dependent variable: Average daily weight gain (ADG, g/day) between 12 to 28 weeks of age.

Hypotheses:
Ho: µA = µB
Ha: µA ≠ µB.

Statistical Test: Two samples

independent t-test
Significance level: .05

94
Q

We can check if there is no outliers in independent t-test by looking at

A

boxplots - no outlier here

95
Q

To check normality of distribution for both independent groups for two-samples independent t-test, we can use..

A

histogram, q-qplot and tests of normality

96
Q

Checking normality for independent, is it - (3)

Research question: Which of the two diet formulas is better for puppies?

Dependent variable: Average daily weight gain (ADG, g/day)

A

We don’t have sig values for either group in the test of normality, histogram and plots look normal

So we have normality of distribution for both independent groups

Inspection of Q-Q Plots and the non-significant Shapiro-Wilk tests (p > .05) indicate that the ADG is normally distributed for both groups

97
Q

. If you find that either one or both of your group’s data is not approximately normally distributed and groups sizes differ greatly, you have two options in independent/paired - (2)

A

1) transform your data so that the data becomes normally distributed,

or (2) run the Mann-Whitney U test which is a non-parametric test that does not require the assumption of normality.

98
Q

For checking homogeneity of variances in independent/paired we use

A

levene’s test

99
Q

Checking homogenity of variance in this two-sample independent t-test, what does it show?

Research question: Which of the two diet formulas is better for puppies?

Dependent variable: Average daily weight gain (ADG, g/day)

A

was homogeneity of variance as assessed by Levene’s Test for Equality of Variances (F = 1.58, p = .219)

100
Q

What does results of two-sample independent results show?

Research question: Which of the two diet formulas is better for puppies?

Dependent variable: Average daily weight gain (ADG, g/day)

A

This study found that puppies in diet B had statistically significantly higher average daily weight gain (89.29 ± 9.93 g/day) between 12 and 28 weeks of age compared to puppies in diet A (60.20 ± 6.85 g/day), t(27)= -9.24, p < .001.

101
Q

In Cohen’s D theoretically 3 SDs can be used - (3) which make very little difference

A
  1. Pooled SD (over conditions)
    2.Averaged SD
  2. Control group SD
102
Q

To calculate Cohen D for independent/paired t-test we need to use

A

control group SD

103
Q

How to calculate Cohen’s D for independent two samples t test for this group?

Research question: Which of the two diet formulas is better for puppies?

Dependent variable: Average daily weight gain (ADG, g/day) - (2)

A

d = (89.29 - 60.20) / 6.85

d = 4.25

104
Q

Cohen’s D guidelines for small, medium large - (3)

A

d = 0.2 be considered a ‘small’ effect size,

d = 0.5 represents a ‘medium’ effect size

d = 0.8 a ‘large’ effect size

105
Q

What is categorical variable? - (2)

A

nominal like unordered e.g., male/female

ordinal e.g., minimal/moderate/severe pain

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