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statistics 10
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- 1assumptions we make for Anova and regression> (Independence) All measurements of each variable were independent of one another
> (Normality) The residuals are normally distributed
> (Homoscedasticity) Spread of observed scores is equal for any predicted value - 24 th assumption for linear regressionLinear relationship between all predictors and the dependent variable (Linearity)
- 3Independence of measurementswe will always assume to have exactly one
measurement of each person for each variable. - 4A violation of independence (plus example)> if one person (secretly) fills in more than one questionnaire, or in some way influences other people
> we all have to take exams for a course at the same time. it was unfair if exams were taken one-by-one in alphabetical order because some peolple would have more time to study.
> makes drawing conclusions
very hard if a group is relatively overrepresented in the first part of the alphabet.
- If, by coincidence,
most guys have their last name come early in the alphabet, you draw the conclusion that girls did better than guys on average, even though the difference in results may have been due to the guys being disadvantaged by having to make the exam earlier. - 5The main method of trying to prevent lack of independence> setting up the research properly.
> After the data is collected, it becomes quite hard to make a distinction between biased scores and legitimate scores
> if source of bias is simply one participant telling one other participant a few correct answers on said exam, it is nearly impossible to determine that this happened by simply looking at
the dataset. - 6Normality assumption in regression is related to...> Normality very specifically about the residuals
> assumption is not about normality of the independent variable or the dependent
variable! - 7to what will normality of both the independent variable and dependent variable will virtually always lead? Other way around?> normality of both the independent variable and dependent variable will virtually always lead to normally distributed residuals
> but the opposite is not always true - 8How to check for normality> histogram
> PP-plot (desined for this assumption)
> QQ-plots (desined for this assumption)
> Boxplots are also useful, especially for ANOVA where each box represents a group, but not they make it harder to look for normality. A perfectly normally distributed boxplot would be both
symmetrical, and have tails that are longer than the center - 9When to reject assumption for normality?> no universal, absolute rule
> look into the data!
> we may also compute statistical tests to determine whether we should reject or not
reject the assumption of normality
- The null hypothesis for any statistical test for normality is therefore that normality is absolutely perfect. - 10Is significance testing for testing normality good?> It is a dangerous game.
>Through the central limit theorem, we know that as the sample size gets larger, normality becomes less of a
problem.
> However, significance tests are also more likely to be significant when the sample size increases. - 11two statistics that can really help us describe the non-normality in a distribution> Skewness & Kurtosis.
> An exact normal distribution will have a skewness and kurtosis of 0. - 12Homoscedasticity of residuals means...that the spread of the observed scores is equal for every level
of the predicted scores, meaning that the residuals have about the same values for entire model. - 13ways to check for homoscedasticity for ANOVA> simply look at the
standard deviations of the separate groups of the independent variables.
> There is a rule of thumb
that the largest standard deviation of said groups should be less than twice as large as the smallest standard deviation. ---> not relying on sample size - 14Levene’s test> significance test for homoscedasticity (only for anova)
> The null hypothesis is that the assumption, in this case homoscedasticity, is perfectly met, so that all groups have exactly the same standard deviation. - 15inspection of homoscedasticity that works for both ANOVA and Linear regression> residual plot
> has the predicted values on the x-axis, and the residuals on the y-axis - 16Violations of homoscedasticity can occur in two distinct ways:> there are outliers,
disproportionally increasing the standard deviation of certain predicted values (or groups) over others
> clear pattern in that the spread of the residuals grows or diminishes for increasing predicted values - 17Whenever more than one continuous variable is present in a single model...> these variables somehow need to be related to each other, and in the end correlation will always be used
- 18when is the linearity assumption not rejected ?> when there is completely no
relationship between two continuous variables.
> In that case, a correlation of 0 adequately reflects
the fact that there is no relationship between the two variables. - 19Why is linearity not of interest for anova?> Because ANOVA only has one continuous variable, which is the dependent variable, there is
no second continuous variable to calculate correlations for.
> Because no correlations are calculated,
there is no risk of them being inadequate, so linearity is not an assumption - 20checking linearity> check this assumption visually
> (Partial plots) and the same residual plots from homoscedasticity
> - 21two ways to deal with lack of linearity> apply a non-linear transformation to the data to create a more linear relationship.
> If the lack of linearity is caused by having variables that are not intrinsically continuous, we may use statistical tests that are explicitly made for this data type. - 22What is the main way to make sure there is no violation of the assumption of independence?A good research design that takes into account potential causes of lack of independence.
- 23Where is the peak of a distribution with a positive skewness and positive kurtosis, compared to an exact normal distribution? (Above/below, right/left)Positive kurtosis means that the peak is too high, so the peak is above an exact normal distribution.
Positive skewness means that there are too many values too far on the right compared to the left, so
the peak is to the left of the exact normal distribution. That means that the peak when they are both
positive is above-left. - 24Why is it a problem for the p-value if the pooled standard deviation is inaccurate for some groups?The problem mainly arises when there is one group with a disproportionately high proportion of the
sample size. Say there is one large group of n = 60 that has an SD of 10, while two other smaller
groups of n = 10 have an SD of 2. The pooled standard deviation would then be about 8.7, which is
relatively accurate for the large group, but a horrible estimation for the smaller groups. In this case,
the SD of 8.7 would also be applied to the smaller groups, meaning that while they may actually
significantly differ from the large group, we might not find this due to the artificially large standard
deviation. The reverse can also happen when it is the large group that has a small standard deviation. - 25Which of the ways to check normality (graph, test, skewness/kurtosis) would you use for a sample size of 100, and why?go visual with the PP-plot and QQ-plot. Skewness and Kurtosis are nice as descriptive statistics. When they are close to 0, they provide a relatively objective measure that normality is fine.
However, if they are not close to 0, there is no way to tell if this is due to a structural deviation from
normality, or due to an outlier messing up the results. The same is true for significance tests. The
main advantage of these is that they provide an extremely accessible p-value, which is useful to relay
the results to beginners. However, unlike the other options, significance tests don’t work properly for
extremely small or large samples. A sample of 100, however, is right in the sweet spot of when it
works appropriately. Finally, graphs are the only ones to clearly show outliers if they are there and is
not dependent on sample size, but looking at patterns is the most subjective out of all options. Still,
with enough experience, this should not be a problem. - 26When we find that every predictor in our linear regression model is statistically significant, proving that the linear relationships are already significant, do we still need to worry about the assumption of linearity, and why (not)?Yes, and this is true for any assumption. The significance of predictors or effects is unrelated to the
assumptions, so they should always be checked. - 27If data for oneway-ANOVA has outliers that we do not wish to remove as they are a crucial component of the data, is it more appropriate to use Kruskal-Wallis rather than oneway-ANOVA, or is it more appropriate the transform the data to try and improve on the assumptions of homoscedasticity and normality?Transformation does not work well with outliers, because even if they could be made to be relatively
closer to the rest of the data, it is often impossible to transform outliers away. For Kruskal-Wallis,
there is no assumed distribution, and any score is converted to a rank score, instantly solving the
problem with outliers. In this case, going with the Kruskal-Wallis test is more appropriate. - 28How does a violation of the assumption of linearity usually affect the p-value?Violation of linearity usually underestimates the effect size, so the p-value will often be too high.
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- Psychologie Fichas sobre statistics 10, creado por Adrian Mayer el 26/11/2018.
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- Adrian Mayer
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