Download - Probability models- the Normal especially
• probability models- the Normal especially
• checking distributional assumptions
Histogram of FS
SEPA location code: 4556FS/100ml
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Normal Q-Q Plot
Theoretical Quantiles
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Histogram of log10(FS)
SEPA location code: 4556log10(FS)/100ml
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Normal Q-Q Plot
Theoretical Quantiles
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FS: Site 9320
Theoretical Percentile (log10 scale): 1.47x
Fn
(x)
1.47 1.75
Theoretical Percentile
Empirical Percentile
Log scale 1.47 1.75
Directive Scale
29.5 56.2
Modelling Continuous Variables checking normality
• Normal probability plot
• Should show a straight line
• p-value of test is also reported (null: data are Normally distributed)C1
Perc
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43210-1-2-3
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Mean
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0.1211StDev 1.015N 100AD 0.361P-Value
Probability Plot of C1Normal
• another statistic- the estimated standard error
Statistical inference
• Confidence intervals
• Hypothesis testing and the p-value
• Statistical significance vs real-world importance
• a formal statistical procedure- confidence intervals
Confidence intervals- an alternative to hypothesis testing
• A confidence interval is a range of credible values for the population parameter. The confidence coefficient is the percentage of times that the method will in the long run capture the true population parameter.
• A common form is sample estimator 2* estimated standard error
• another formal inferential procedure- hypothesis testing
Hypothesis Testing
• Null hypothesis: usually ‘no effect’
• Alternative hypothesis: ‘effect’
• Make a decision based on the evidence (the data)
• There is a risk of getting it wrong!
• Two types of error:-– reject null when we shouldn’t
- Type I– don’t reject null when we should
- Type II
Significance Levels
• We cannot reduce probabilities of both Type I and Type II errors to zero.
• So we control the probability of a Type I error.
• This is referred to as the Significance Level or p-value.
• Generally p-value of <0.05 is considered a reasonable risk of a Type I error.(beyond reasonable doubt)
Statistical Significance vs. Practical Importance
• Statistical significance is concerned with the ability to discriminate between treatments given the background variation.
• Practical importance relates to the scientific domain and is concerned with scientific discovery and explanation.
Power
Power is related to Type II error
probability of power = 1 - making a Type II
error
Aim:
to keep power as high as possible
Statistical models
• Outcomes or Responsesthese are the results of the practical work and are sometimes referred to as ‘dependent variables’.
• Causes or Explanationsthese are the conditions or environment within which the outcomes or responses have been observed and are sometimes referred to as ‘independent variables’, but more commonly known as covariates.
• relationships- linear or otherwise
Correlations and linear relationships
• pearson correlation
• Strength of linear relationship
• Simple indicator lying between –1 and +1
• Check your plots for linearity
gene correlations
1.11.00.90.80.70.60.50.4
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0.90.80.70.60.50.4
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Interpreting correlations
• The correlation coefficient is used as a measure of the linear relationship between two variables,
• The correlation coefficient is a measure of the strength of the linear association between two variables. If the relationship is non-linear, the coefficient can still be evaluated and may appear sensible, so beware- plot the data first.
A matrix plot
1209060 1.00.50.0 16808.8
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pH (pH units)
O2 -%sat (%)
BOD (ATU) (mg/ L)
Ammonia as N (mg/ L)
o-Phos as P (mg/ L)
Fe (mg/ L)
Matrix Plot of pH (pH units, O2 -% sat (% ), BOD (ATU) (m, ...
181614121086420
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Fe (mg/ L)
Am
monia
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mg/L)
Scatterplot of Ammonia as N (mg/ L) vs Fe (mg/ L)
181614121086420
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Fe (mg/ L)
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Scatterplot of o-Phos as P (mg/ L) vs Fe (mg/ L)
1.00.80.60.40.20.0
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Ammonia as N (mg/ L)
o-P
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Scatterplot of o-Phos as P (mg/ L) vs Ammonia as N (mg/ L)
Correlations
• P and N, 0.228 (p-value 0.001)
• Fe and N, 0.174 (p-value 0.008)
• Fe and P, 0.605 (p-value 0.000)
• all highly significant, but do the scatterplots support this interpretation?
• points tend to be clustered in bottom left corner of plot,
• there are one or two observations well separated from the cluster
• both might suggest a transformation (try logs)
3210-1
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log Fe
log P
Scatterplot of log P vs log Fe
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log Fe
log N
Scatterplot of log N vs log Fe
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log N
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Scatterplot of log P vs log N
Correlations
• logP, logN 0.167 (p-value 0.012)
• logFe, LogN 0.134 (p-value 0.043)
• logP, log Fe, 0.380 (p-value 0.000)
• what is a statistical model?
Statistical models
• In experiments many of the covariates have been determined by the experimenter but some may be aspects that the experimenter has no control over but that are relevant to the outcomes or responses.
• In observational studies, these are usually not under the control of the experimenter but are recorded as possible explanations of the outcomes or responses.
Specifying a statistical models
• Models specify the way in which outcomes and causes link together, eg.
• Metabolite = Temperature• The = sign does not indicate equality in a
mathematical sense and there should be an additional item on the right hand side giving a formula:-
• Metabolite = Temperature + Error
statistical model interpretation
• Metabolite = Temperature + Error
• The outcome Metabolite is explained by Temperature and other things that we have not recorded which we call Error.
• The task that we then have in terms of data analysis is simply to find out if the effect that Temperature has is ‘large’ in comparison to that which Error has so that we can say whether or not the Metabolite that we observe is explained by Temperature.
summary
• hypothesis tests and confidence intervals are used to make inferences
• we build statistical models to explore relationships and explain variation
• the modelling framework is a general one – general linear models, generalised additive models
• assumptions should be checked.