Data Screening and T-tests

Unit Learning Outcomes

• Describe, summarise, and appropriately present data.
• Screen and appropriately transform data.
• Select appropriate statistical analysis methods for data sets and perform these procedures using statistical software.
• Interpret, evaluate, and describe the output of statistical analyses in a manner appropriate for a scientific report.
• Interpret the outcomes of published research trials.

Week 2 Learning Outcomes

• Parametric and Non-parametric tests
• Normality Testing - Transformations
• t-Tests
  • one-sample t-test
  • independent t-tests
  • dependent/paired t-tests
• Recognise when to use t-tests
• Perform, interpret, and present the results of t-tests

Quick Revision

• Hypotheses
  • Null Hypothesis
  • Alternate Hypothesis
• ‘P’ value

Formulating a Hypothesis

• Hypothesis of No Difference (H0) – "Null hypothesis"
  • Example: HO: There is no significant difference in mean soil temperature between the sun and the shade.
  • Example: HO: The distribution of our data is not significantly different to the Normal Distribution
  • These are often implied when you formulate your hypotheses or when you read a scientific paper.
• Hypothesis of a Difference (HA) – the "alternative hypothesis"
  • Example: HA: Mean soil temperature in the sun and the shade are significantly different.
  • Example: HA: Mean heart rate of subjects given drug A and drug B are significantly different.
  • Example: HA: The mean heights of sugar cane plants on three fertiliser regimes (100kg/ha, 200kg/ha and 300 kg/ha) are all not the same.

Probability Level for Claiming a Result is “Significant

• In science, the significance level α is set at 5% or 0.05.
  • We reject the null hypothesis when the probability that it might be true is less than 5% or 0.05.
  • If our ‘p’ value is less than 0.05, we are more than 95% sure that we could accept the alternate hypothesis.

How to Accept or Reject HO?

• All our statistical tests:
  • Calculate a test statistic (t) from our samples and compare this to a table of critical values (for α = 0.05)
  • if t ≥ critical value then reject H0
  • Or most commonly these days the statistical software will give you a p-value

P-value

• A p-value is the probability of achieving the calculated test statistic or a larger test statistic given the assumed distribution.
• α is the ‘cut-off’ point to decide whether an event ‘really’ happened or not.
• Reject H0 if p < 0.05
• A p-value is the probability that the null hypothesis is true.

Parametric Tests

• A test that makes assumptions about the parameters of the population distribution from which it is drawn.
  • Data must conform to the assumptions (e.g. normally distributed, homogeneity of variances)
• Parametric tests include: t-tests, ANOVA’s, regression and correlation.
• Data ideally should conform to this. In reality, biological studies often don’t, and we just deal with that.

Non-parametric Tests

• A test that makes no assumptions about the parameters of the population distribution from which it is drawn.
• Used when the data do not conform to the assumptions of parametric tests (e.g. normal distribution, homogeneity of variance).
• Non-parametric tests include: Chi-square, contingency tables, Spearman-Rank correlation, Mann-Whitney and Kruskal-Wallis tests.
  • The trade-off here of having no assumptions is reduced sensitivity.

Data Screening

• Testing for normality and homogeneity of variance.
• What to do if your data fail these tests!!!!

Normality

• The data should be normally distributed BUT many statistical tests are nevertheless robust to deviations.
• Testing for normality
  • Numerical - Shapiro-Wilks, Kolmogorov- Smirnov etc.
  • Graphical - histograms, boxplots, QQ plots, normal probability plots, detrended normal plot, etc

Hypotheses for Normality Tests

• Tested at ($\alpha = 0.05$)
• H0: The data come from a population which is not different to a normal distribution
• HA: The data come from a population which is different to a normal distribution

Graphical Tests of Normality - Histograms

• Relatively Normal
• Positive Skew
• Negative Skew
• Bi-Modal

Graphical Tests of Normality - Boxplots

• Extreme value
• Median
• minimum
• maximum
• 75th Percentile
• 25th Percentile

Graphical Tests of Normality – Normal Probability Plots (QQ Plots)

• Normal
• Positive Skew
  • Fat positive tail
  • Positive skew (A greater frequency of large measurements than expected)
  • Middle slightly negative
• Negative Skew
  • Thicc negative tail
  • Negative skew (a greater frequency of small measurements than expected)

Homogeneity of Variance

• The variances of all treatment groups should be similar.
• Homogeneity of variance is especially important with unequal group sizes.
• Beware unequal and/or small (< 6 per group) sample sizes. These usually do NOT have homogenous variance.
• Variance = SD$\^2$

Hypotheses for Homogeneity of Variance (Levene’s Test)

• ($\alpha = 0.05$)
• H0: There is no difference in the amount of variance between treatment groups. (i.e. have equal variances, homogenous variance).
• HA: The variances of the treatment groups are different (i.e. heterogeneous variance).

Levene’s Test

• Levene’s test is a robust test as it handles data sets that deviate from a normal distribution.
• Hypotheses for Levene’s Test: tested at ($\alpha = 0.05$)
• H0: The treatment groups have equal variances (i.e. homogenous variance).
• HA: The variances of the treatment groups are not all the same (ie heterogeneous variance).
• P > 0.05 so ACCEPT HO

The Importance of Meeting Assumptions

• Homogeneity of variance is a more important assumption than normality
  • i.e. some deviation from Normality will not affect results as much as non-homogeneous variance

Data Transformation

• Why transform?
• How do you transform data?
• Choosing the right transformation.

Why Transform? – Normalise the Data!

Data Transformation - Common Transformations

• x = each value in the data
• K = your largest value + 1
• C = is a constant you add to make your smallest value 1 (usually 1)

How to Transform Data

• e.g. log 10 transformation
  • log10 (55) =
  • log10 (48) =

How to Transform Data - Examples of Transformations

When to Transform?

• Remember - Transformations on small data sets are rarely very successful.
• When p < 0.05 for normality and homogeneity of variance tests AND difference is visible in plots.
• BUT – what if only 1 data vector has a p<0.05?
• MUST transform all data vectors that you are comparing
• Once transformed – carry out statistical analysis USING transformed data
• Report – Test statistic, df and p value
• Remember to use original numbers (‘back-transform’) for graphs / interpretation – make sense of the results.

T-Tests

• t-tests to compare the means of two groups of data to establish whether they are different or not.
• When are they used?

Data Assumptions for t-tests

• t-tests are parametric tests and therefore have a number of data assumptions:
  1. Normal Distribution
  2. Homogeneity of variance
  3. Random assignment of subjects
• If these are violated, it is not appropriate to perform a t-test.
  • Instead, it is necessary to perform a non-parametric test (covered later in the course).

Three Types of t-Tests

1. One-sample
2. Independent
3. Dependent / paired t-tests

One-Sample t-Test

• Compare the mean of your samples with an expected mean
• Example: A scientist measured the body temperature of a bunch of crabs and wanted to know if their mean body temperature tended to ambient after exposure for a period of time. Ambient temperature was 24.30C.
  • H0: µ = 24.30C
  • HA: µ ≠ 24.30C
  • H0: The body temperature of this sample of crabs is not significantly different from ambient (24.30C)
  • HA: The body temperature of this sample of crabs is significantly different from ambient (24.3 0C)

Two Samples – Independent t-Test

• Testing for differences between the means of both samples
• Example: In order to deduce the relative intelligence of two species, an anthropologist is investigating the brain capacity of hominoids in the genus Homo.
  • H0: µ(Hh) = µ(He)
  • HA: µ(Hh) ≠ µ(He)
  • H0: The mean brain capacity of Homo habilis and Homo erectus are not different
  • HA: The mean brain capacity of Homo habilis and Homo erectus are significantly different

Two Samples – Dependent (Paired) t-Test

• Testing for differences between the means of repeated or paired measurements on the same samples
• Example: As a test for stress, blood cortisol levels (ng/ml) were measured in 8 wombats when they were newly captured and again after one month in captivity.
  • H0: µ(Time 1) = µ(Time 2)
  • HA: µ(Time 1) ≠ µ(Time 2)
  • H0: Mean blood cortisol levels are not different between Time 1 and Time 2
  • HA: Mean blood cortisol levels are significantly different between Time 1 and Time 2

One-Sample t-Test Explained

• Measures if the sample mean from one set of measurements is different from the reference mean
• Where:
  • Population mean
  • Sample mean
  • Std error
• Hypotheses (tested at α = 0.05):
  • HO: µ1 = µ2 HA: µ1 ≠ µ2
  • HO: There is no significant difference between the sample mean and test mean
  • HA: The sample mean and test mean are not the same
  • Reject HO if: If t(calculated) > t(critical) (or) p < 0.05

One-Sample t-Test - Worked Example

• Do quokkas from the mainland (Pinjarra) have a similar body size to those found on Rottnest Island?
  • Adult male quokkas from Rottnest average 22.8 cm. Adult males from the mainland colony have the following body length measurements (cm):
  • H0: x = 22.8 cm
  • HA: x ≠ 22.8 cm
  • H0: There is no significant difference in mean body length between the Rottnest Island and mainland Quokka colonies
  • HA: There is a significant difference in mean body length between the Rottnest Island and mainland Quokka colonies
• Rstudio output:
  • p > 0.05 therefore accept HO

Results of One-Sample t-Test Example

• A one sample t-test ($\alpha$ = 0.05) was performed to determine whether the mean body length of adult male quokkas from two colonies (Rottnest and Pinjarra) were similar. There was no significant difference in mean length between colonies OR Quokkas from both colonies were similar in mean length (t = 0.246; df = 6; p = 0.814).
• The mean body lengths of Rottnest and Pinjarra quokkas were 22.8 cm and 22.6 cm (± 0.99s.e.) respectively.

Worked Example (using T Scores)

• Longevity of a sample of 9 Humans
  • Population norms: mean = 80, S.D = 10 Sample: mean = 85, n=9, $\alpha$ = 0.05 (probability level)
• CALCULATIONS
  1. Sample mean (85) – population mean (80) = 5
  2. S.D (10)/ SQRT (9) = 3.334
  3. t value = 5/3.334 = 1.5
• Degrees of freedom (f) = n - 1 (for one sample & dependent)
  • n - 2 (for independent samples)

Two-Sample Independent t-Test

• t = difference between group means / variability of groups
• Measures the probability of overlap between the distributions of 2 means
• Equation:
  • t = XA-XB / SEd
• Two sources of variance when looking at groups of data:
  1. between the groups and
  2. within the groups
• Big variance BETWEEN groups compared with WITHIN groups = Significant difference between groups = REJECT HO
• Big variance WITHIN groups compared with BETWEEN groups = NO significant difference between groups = ACCEPT HO

How to Reject/Accept Hypothesis

• Either:
  • p<0.05 reject HO (if using RStudio or any computer package)
• Or:
  • compare tcalculated to tcritical (from tables) to judge significance. If tcalculated > tcritical reject HO

Two-Sample Independent t-Test - Worked Example 1

• In order to deduce the relative intelligence of two species, an anthropologist is investigating the brain capacity of hominoids in the genus Homo.
  • An independent t-test indicated that the mean brain capacity (cm3)of Homo habilis and Homo erectus are not the same (t = 11.06; df = 12; p < 0.001). The mean brain capacity of Homo habilis and Homo erectus were 656 and 894 cm3 respectively .
• Discussion/conclusions: If brain capacity is indicative of intelligence, Homo erectus was more intelligent that Homo habilis.

Two-Sample Independent t-Test - Worked Example 2

• A forensic scientist wants to determine whether two cannabis samples have come from the same supplier. Measures the THC content of cannabis in 2 samples (%w/v) (n=7)
• RESULTS: An independent t-test (n = 7; α = 0.05) indicated that the mean THC content (%w/v) of cannabis from sample one and sample two were not statistically different (t = 0.16; df = 12; p = 0.875). The mean %w/v of the samples were 4.043 and 4.057 for sample one and sample two respectively.

Dependent (Paired) t-Test

• To test whether the means of two sets of paired measurements are different from each other.
• Look at the differences between each pair of points and then see if the mean of these values is significantly different from zero.
  • using a one-sample t-test with a comparison mean of zero
• How to use the formula: Calculate the t value and use the tables to reject/accept the null hypothesis
• DF = n-1
• S.E = SD ÷ √n
• T = mean diff ÷ S.E. diff

Independent Versus Paired Test

• Paired t-Test – does not consider within group variability – therefore it is a more sensitive test
  • More likely to be significant
• Independent – considers within group variability – a penalty factor which results in a larger SE and smaller t value (Look at equation!)
  • Smaller T-value = less likely to be significant

Review Question

• The water content of a sample of nine sediment cores taken at random is: 6.1, 5.5, 5.3, 6.8, 7.6, 5.3, 6.9, 6.1, 5.7 wt% H2O
  • H0: The sample is from a population with a mean content of 7.0% H2O
  • HA: The sample is from a population with a mean water content that is NOT 7.0% H2O
  • SD = 0.803, α=0.05 and 0.01
• Calculate which hypothesis is true for each α value