SDA_M1_Sess3

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Session 8
Tests of Hypotheses
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Learning Objectives

• By the end of this session, you will be able to
• set up, conduct and interpret results from a test
  of hypothesis concerning a population mean
• explain how means from two populations may be
  compared, and state assumptions associated with
  the independent samples t-test
• interpret computer output from one or two-sample
  t-tests, present and write up conclusions
  resulting from such tests
• explain the difference between statistical
  significance and an important result

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An illustrative example

• Farmers growing maize in a certain area were
  getting average yields of 2900 kg/ha.
• A new Integrated Pest Management (IPM) approach
  was attempted with 16 farmers.
• Objective To determine if the new approach
  results in an increase in maize yields.
• Yields from these 16 farmers (after using IPM)
  gave mean 3454 kg/ha,
• with standard deviation 672 kg/ha hence s.e.
  168.
• Can we determine whether IPM has really increased
  maize yields?

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Is the yield increase real?

• In above example, clearly the sample mean of 3454
  kg/ha is greater than 2990 kg/ha
• But the question of interest is
• does this result indicate a significant increase
  in the yield or might it just be a result of the
  usual random variation of yield
• Hypothesis testing seeks to answer such questions
  
• by looking at the observed change relative to the
  noise, i.e. the standard error in the sample
  estimate

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Null H0 Alternative H1

• Null hypothesis H0 ? 2900
• where ? is the true mean yield of farmers in the
  area using the new approach
• The promoters of the new approach are confident
  that yields with the new approach cannot possibly
  decrease.
• Hence the above null hypothesis needs to be
  tested against the alternative hypothesis
• H1 ? gt 2900

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Testing the hypothesis
Compute the t test statistic t ( -
?)/(s/?n) (3454 2900)/(168) 3.30 which
follows a t-distribution with n-115 degrees of
freedom. Use values of the t-distribution to
find the probability of getting a result, which
is as extreme, or more extreme than the one
(3.30) observed, given H0 is true. The smaller
this probability value, the greater is the
evidence against the null hypothesis. This
probability is called the p-value or significance
level of the test
   
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Analysis in Stata
Type db ttesti or look for the One-sample mean
comparison calculator on the menu
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Results
t-value
t-probabilities from formulae or table
Result from the one-sided test done here
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Interpretation and conclusions
It is clear from t-tables that the p-value is
smaller than 0.01. Using statistical software, we
get the exact p-value as 0.0024. This p-value is
so small, there is sufficient evidence to reject
H0. Conclusion Use of the new IPM technology
has led to an increase in maize yields
(p-value0.0024)
   
   
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An example Comparing 2 means
As part of a health survey, cholesterol levels of
men in a small rural area were measured,
including those working in agriculture and those
employed in non-agricultural work. Aim To see if
mean cholesterol levels were different between
the two groups.
Agric Non-agric
156 223
282 131
222 137
172 146
183 130
206 122
210 141
198 192
199 188
211 212
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Summary statistics
Begin with summarising each column of data.
Agric Non-agric
Mean 203.9 162.2
Std. dev. 33.9 37.6
Variance 1147 1412
There appears to be a substantial difference
between the two means. Our question of interest
is Is this difference showing a real effect, or
could it merely be a chance occurrence?
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Setting up the hypotheses
To answer the question, we set up Null
hypothesis H0 no difference between the two
groups (in terms of mean response), i.e. ?1
?2 Alternative hypothesis H1 there is a
difference, i.e. ?1 ? ?2 The resulting test will
be two-sided since the alternative is not equal
to.
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Test for comparing means

• Use a two-sample (unpaired) t-test
• - appropriate with 2 independent samples
• Assumptions
• - normal distributions for each sample
• - constant variance (so test uses a pooled
  estimate of variance)
• - observations are independent
• Procedure
• - assess how large the difference in means is,
  relative to the noise in this difference, i.e.
  the std. error of the difference.

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Test Statistic
The test statistic is
where s2, the pooled estimate of variance, is
given by
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Numerical Results
The pooled estimate of variance, is
1279.5 Hence the t-statistic is
41.7/?(2x1279.5/10) 2.61 , based
on 18 d.f. Comparing with tables of t18, this
result is significant at the 2 level, so reject
H0. Note The exact p-value 0.018
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Results and conclusions
Difference of means 41.7 Standard error of
difference 15.99 95 confidence interval for
difference inmeans (8.09, 75.3). Conclusions
There is some evidence (p0.018) that the mean
cholesterol levels differ between those working
in agriculture and others. The difference in
means is 42 mg/dL with 95 confidence interval
(8.1, 75.3).
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Analysis in Stata
Input the data and do a t-test Or complete the
dialogue as shown below Or type ttesti 10 203.9
33.9 10 162.2 37.6
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Results
This was a 2-sided test
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General reporting the results

• Take care to report results according to size of
  p-value.
• For example, evidence of an effect is
• almost conclusive if p-value lt 0.001 and could
  be said to be strong if p-value lt 0.010
• If 0.01lt p-value lt 0.05, results indicate some
  evidence of an effect.
• If p-value gt 0.05, but close to 0.05, it may
  indicate something is going on, but further
  confirmatory study is needed.

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Significance further comments

• e.g. Farmers report that using a
  fungicideincreased crop yields by 2.7 kg ha-1,
  s.e.m.0.41
• This gave a t-statistic of 6.6 (p-valuelt0.001)
• Recall that the p-value is the probability of
  rejecting the null hypothesis when it is true.
• i.e. it is the chance of error in your conclusion
  that there is an effect due to fungicide!

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How important are sig. tests?

• In relation to the example on the previous
  slide,we may find one of the following
  situations fordifferent crops.
• Mean yields
• with and without fungicide.
• 589.9 587.2 ? Not an important
  finding!
• 9.9 7.2 ? Very important finding!
• It is likely that in the first of these results,
  either too much replication or the incorrect
  level of replication had been used (e.g. plant
  level variation, rather than plot level variation
  used to compare means).

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What does non-significance tell us?

• e.g. There was insufficient evidence in the data
  todemonstrate that using a fungicide had any
  effecton plant yields (p0.128).
• Mean yields with and without fungicide.
• 157.2 89.9
• This difference may be an important finding, but
  thestatistical analysis was unable to pick up
  this differenceas being statistically
  significant.
• HOW CAN THIS HAPPEN?
• Too small a sample size? High variability in the
  experimental material? One or two outliers? All
  sources of variability not identified?

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Significance Key Points

• Statistical significance alone is not enough.
  Consider whether the result is also
  scientifically meaningful and important.
• When a significant result if found, report the
  finding in terms of the corresponding estimates,
  their standard errors and C.I.s
• (as is done by Stata)