inferential statistics inferential statistics: the part of statistics that allows researchers to...

Inferential Statistics

Inferential statistics: The part of statistics that allows researchers to generalize their findings beyond data collected.

Statistical inference: a procedure for making inferences or generalizations about a larger population from a sample of that population

Research is about trying to make valid inferences

How Statistical Inference Works

Basic Terminology Population (statistical population):

Any collection of entities that have at least one characteristic in commonA collection (a aggregate) of measurement about which an inference is desiredEEverything you wish toverything you wish to studystudy

Parameter: The numbers that describe characteristics of scores in the population (mean, variance, standard deviation, correlation coefficient etc.)

N = 28 N = 28 μμ = 44 = 44 σσ² = 1.214² = 1.214

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45A Population of Values

Body Weight Data (Kg)Population

Basic Terminology Sample:

A part of the populationA finite number of measurements chosen from a population

Statistics: The numbers that describe characteristics of scores in the sample (mean, variance, standard deviation, correlation coefficient, reliability coefficient, etc.)

nn = = 11 value value … …

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Body Weight Data (Kg)

X1: 43

X: student body weight

nn = = 22 values values … …

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x: student body weight

5 values5 values … …

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x1: 43 x2: 44 x3: 45 x4: 44x5: 44

a sample that has been selected in such a way that all members of the population have an

equal chance of being picked (A Simple Random Sample )

Basic concept of statistics Measures of central Measures of central tendency

Measures of dispersion & variability

Measures of tendency centralMeasures of tendency centralArithmetic mean (= simple average)

summationmeasurement in population

index of measurement

• Best estimate of population mean is the sample mean, X

n

XX

n

ii

1sample size

Measures of variabilityMeasures of variabilityAll describe how “spread out” the dataAll describe how “spread out” the data

1. Sum of squares,sum of squared deviations from the mean

• For a sample,

2)( XXSS i

2.2. Average or mean sum of Average or mean sum of squares = variance, squares = variance, ss22::

• For a sample,

12

2

n

XXs i )(

Why?

nn – 1 represents the – 1 represents the degrees of degrees of freedomfreedom, , , or number of independent , or number of independent quantities in the estimate quantities in the estimate ss22..

12

2

n

XXs i )(

• therefore, once n – 1 of all deviations are specified, the last deviation is already determined.

01

n

ii XX )(Greek

letter “nu”

3.3. Standard deviation, Standard deviation, ss

• For a sample, 12

n

XXs i )(

• Variance has squared measurement units – to regain original units, take the square root

4.4. Standard error of the meanStandard error of the mean

• For a sample,nssX

2

Standard error of the mean is a measure of variability among the means of

repeated samples from a population.

Basic Statistical Symbols

N = 28 N = 28 μμ = 44 = 44 σσ² = 1.214² = 1.214

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Body Weight Data (Kg)Population

repeated random samplingrepeated random sampling, each with sample size, , each with sample size, nn = 5 values = 5 values

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A Population of ValuesBody Weight Data (Kg)

Repeated random samples, Repeated random samples, each with sample size, each with sample size, nn = 5 values = 5 values … …

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SummarySample Samplin

g 1Samplin

g 2Samplin

g 2First 43 (-1) 46 (+1) 42 (-1)Second 44 (+0) 44 (-1) 42 (-1)Third 45 (+1) 46 (+1) 43 (+0)Fourth 44 (+0) 45 (+0) 45 (+2)Fifth 44 (+0) 44 (-1) 43 (+0)

Average 44 45 43Sum of square

2 4 6

Mean square

0.50 1.00 1.50

Standard deviation

0.707 1.00 1.225

For a large enough number of large For a large enough number of large samples, the frequency distribution samples, the frequency distribution of the sample means (= sampling of the sample means (= sampling

distribution), approaches a normal distribution), approaches a normal distribution.distribution.

Sample mean

Freq

uenc

y

Normal distribution: bell-shaped curveNormal distribution: bell-shaped curve

Testing statistical hypothesesTesting statistical hypotheses between 2 means between 2 means

1.1. State the research question in State the research question in terms of statistical hypotheses.terms of statistical hypotheses.It is always started with a statement that hypothesizes “no difference”, called the null hypothesis = H0.

H0: Mean heightof female student is equal to mean height of male student

Then we formulate a statement that Then we formulate a statement that must be true if the null hypothesis must be true if the null hypothesis is false, called the is false, called the alternate alternate hypothesishypothesis = = HHAA . .HA: Mean height of female student

is not equal to mean height of male student

If we reject H0 as a result of sample evidence, then we conclude that HA

is true.

2. Choose an appropriate statistical test that would allow you to reject H0 if H0 were false. E.g., Student’s E.g., Student’s tt test for hypotheses test for hypotheses about meansabout means

William Sealey Gosset

(“Student”)

21

21

XXsXXt

Standard error of the difference

between the sample means

To estimate s(X1 - X2), we must first

know

the relation between both populations.

Mean of sample 2

Mean of sample 1

t Statistic,

How to evaluate the success of this experimental design class

Compare the score of statistics and experimental design of several student

Compare the score of experimental design of several student from two serial classes

Compare the score of experimental design of several student from two different

classes

1. Comparing the score of statistics and experimental experimental design of several student

Similar Student

Dependent

populations

Identical Variance

Different Student

Independent

populations

Identical Variance

Not Identical Variance

Different Student

Independent

populations

Identical Variance


2. Comparing the score of experimental design of several student from two serial classes

3. Comparing the score of experimental design of several student from two classes

Different Student

Independent

populations

Identical Variance


Relation between populationsRelation between populations Dependent populations Independent populations

1. Identical (homogenous ) variance

2. Not identical (heterogeneous) variance

Sample Null hypothesis: The mean difference is equal to o

Dependent Populations

Test statistic Null distributiont with n-1 df

*n is the number of pairscompare

How unusual is this test statistic?

P < 0.05 P > 0.05

Reject Ho Fail to reject Ho

t d do

SEd

Pooled variance:Pooled variance:21

222

2112

sssp

Then,

2

2

1

2

21 ns

ns

s ppXX

Independent Population with homogenous variances

t Y 1 Y 2SE

Y 1 Y 2

SEY 1 Y 2 sp

2 1n1

1n2

21

222

2112

dfdfsdfsdfsp

Independent Population with homogenous variances

When sample sizes are small, the sampling distribution is described better by the t distribution than by

the standard normal (Z) distribution.

Shape of t distribution depends on degrees of freedom, = n – 1.

Z = t(=)t(=25)

t(=1)t(=5)

t

t

Area of Rejection

Area of Acceptance

Area of Rejection

Lower critical value

Upper critical value

0

0.95 0.0250.025For = 0.05

The distribution of a test statistic is divided into The distribution of a test statistic is divided into an area of acceptance and an area of rejection.an area of acceptance and an area of rejection.

Critical t for a test about equality = t(2),

t Y 1 Y 2s12

n1

s22

n2

df

s12

n1

s22

n2

2

s12 n1 2n1 1

s22 n2 2n2 1

Independent Population with heterogenous variances

Analysis of VarianceAnalysis of Variance(ANOVA)(ANOVA)

Independent T-testIndependent T-test Compares the means of one variable for TWO

groups of cases. Statistical formula:

Meaning: compare ‘standardized’ mean difference But this is limited to two groups. What if groups

> 2?• Pair wised T Test (previous example)• ANOVA (Analysis of Variance)

21

21

21

XXXX S

XXt

From T Test to ANOVAFrom T Test to ANOVA11. Pairwise T-Test

If you compare three or more groups using t-tests with the usual 0.05 level of significance, you would have to compare each pairs (A to B, A to C, B to C), so the chance of getting the wrong result would be: 1 - (0.95 x 0.95 x 0.95) = 14.3% Multiple T-Tests will increase the false alarm.

2. 2. Analysis of Variance In T-Test, mean difference is used.

Similar, in ANOVA test comparing the observed variance among means is used.

The logic behind ANOVA:• If groups are from the same population,

variance among means will be small (Note that the means from the groups are not exactly the same.)

• If groups are from different population, variance among means will be large.

From T Test to ANOVAFrom T Test to ANOVA

What is ANOVA?What is ANOVA? Analysis of Variance A procedure designed to determine if the

manipulation of one or more independent variables in an experiment has a statistically significant influence on the value of the dependent variable.

Assumption:Each independent variable is categorical

(nominal scale). Independent variables are called Factors and their values are called levels.

The dependent variable is numerical (ratio scale)

What is ANOVA?What is ANOVA?The basic idea of Anova:

The “variance” of the dependent variable given the influence of one or

more independent variables {Expected Sum of Squares for a Factor} is checked to see if it is

significantly greater than the “variance” of the dependent variable

(assuming no influence of the independent variables) {also known as the Mean-Square-Error (MSE)}.

Pair-t-TestAmir 6 Budi 9Abas 8 Berta 4Abi 10 Bambang 7Aura 6 Banu 5Ana 10 Betty 5

Average 8 6n 5 5Var. sample 4 4

Pooled Var. = 4 tcalctcalc =1.581t-tablet-table 2.306

ANOVA TABLE OF 2 POPULATIONS

S V SS DF Mean square (M.S.)

Between populations

Within populations

SSbetween

1 MSBSSBDFB

SSWithin

(n1-1)+ (n2-1)

SSWDFW

= MSW

=

TOTAL SSTotal n1 + n2 -1

S²

ANOVA TABLE OF 2 POPULATIONS

S V SS DF Mean square (M.S.)

Between populations

Within populations

10 1 10

32 8 4

TOTAL 42 9

Fcalc = 2.50

Ftable = 5.318

Rationale for ANOVARationale for ANOVA• We can break the total variance in a study We can break the total variance in a study

into meaningful pieces that correspond to into meaningful pieces that correspond to treatment effects and error. That’s why treatment effects and error. That’s why we call this Analysis of Variance.we call this Analysis of Variance.

GXThe Grand Mean, taken over all observations.

AX

1AX

The mean of any group.

The mean of a specific group (1 in this case).

iXThe observation or raw data for the ith subject.

The ANOVA ModelThe ANOVA Model)()( AiGAGi XXXXXX

Trial i The grand mean

A treatment

effect

Error

SS Total = SS Treatment + SS Error

Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means using data obtained from observational or experimental studies.

Use the sample results to test the following hypotheses.

H0: 1=2=3=. . . = kHa: Not all population means are equal

If H0 is rejected, we cannot conclude that all population means are different.

Rejecting H0 means that at least two population means have different values.

Analysis of VarianceAnalysis of Variance

Assumptions for Analysis of VarianceAssumptions for Analysis of Variance

For each population, the response variable is normally distributed.

The variance of the response variable, denoted 2, is the same for all of the populations.

The effect of independent variable is additive

The observations must be independent.

Analysis of Variance:

Between-Treatments Estimate of Population Variance

Within-Treatments Estimate of Population Variance

Comparing the Variance Estimates: The F Test

ANOVA Table

Testing for the Equality of t Population Means

A between-treatments estimate of σ2 is called the mean square due to treatments (MSTR).

The numerator of MSTR is called the sum of squares due to treatments (SSTR).

The denominator of MSTR represents the degrees of freedom associated with SSTR.

Between-Treatments Estimate Between-Treatments Estimate of Population Varianceof Population Variance

2

1( )

MSTR 1

k

j jj

n x x

k

The estimate of 2 based on the variation of the sample observations within each treatment is called the mean square due to error (MSE).

The numerator of MSE is called the sum of squares due to error (SSE).

The denominator of MSE represents the degrees of freedom associated with SSE.

Within-Treatments Estimate Within-Treatments Estimate of Population Varianceof Population Variance

2

1( 1)

MSE

k

j jj

T

n s

n k

Comparing the Variance Estimates: Comparing the Variance Estimates: The The F F Test Test

If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSTR/MSE is an F distribution with MSTR d.f. equal to k - 1 and MSE d.f. equal to nT - k.

If the means of the k populations are not equal, the value of MSTR/MSE will be inflated because MSTR overestimates σamong2

Hence, we will reject H0 if the resulting value of MSTR/MSE appears to be too large to have been selected at random from the appropriate F distribution.

Test for the Equality of Test for the Equality of kk Population Population MeansMeans

Hypotheses H0: 1=2=3=. . . = k

Ha: Not all population means are equal

Test StatisticF = MSTR/MSE

Test for the Equality of Test for the Equality of kk Population Population MeansMeans

Rejection Rule Using test statistic: Reject H0 if F > Fa

Using p-value: Reject H0 if p-value < a

where the value of Fa is based on an F distribution with t - 1 numerator degrees of freedom and nT - t denominator degrees of freedom

The figure below shows the rejection region associated with a level of significance equal to where F denotes the critical value.

Sampling Distribution of MSTR/MSESampling Distribution of MSTR/MSE

Do Not Reject H0 Reject H0

MSTR/MSE

Critical ValueF

ANOVA TableANOVA TableSource of Sum of Degrees of MeanSource of Sum of Degrees of MeanVariation Squares Freedom Squares FVariation Squares Freedom Squares FTreatmentTreatment SSTRSSTR kk- 1- 1 MSTR MSTR/MSEMSTR MSTR/MSEErrorError SSESSE nnT T - - kMSEMSETotalTotal SSTSST nnTT - 1 - 1

SST divided by its degrees of freedom nT - 1 is simply the overall sample variance that would be obtained if we treated the entire nT observations as one data set.

k

j

n

iij

j

xx1 1

2 SSESSTR)(SST

What does Anova tell us?What does Anova tell us?

ANOVA will tell us whether we have sufficient evidence to say

that measurements from at least one treatment differ significantly

from at least one other.It will not tell us which ones

differ, or how many differ.

ANOVA vs t-testANOVA vs t-test ANOVA is like a t-test among multiple

data sets simultaneously• t-tests can only be done between two data

sets, or between one set and a “true” value

ANOVA uses the F distribution instead of the t-distribution

ANOVA assumes that all of the data sets have equal variances• Use caution on close decisions if they

don’t

ANOVA – a Hypothesis TestANOVA – a Hypothesis Test H0:

There is no significant difference among the results provided by treatments.

Ha: At least one of the treatments provides results significantly different from at least one other.

Yij = + j + ij

By definition, j = 0

t

j=1

The experiment produces

(r x t) Yij data values.

The analysis produces estimates of t

(We can then get estimates of the ij by subtraction).

Linear Model

Y11 Y12 Y13 Y14 Y15 Y16 … Y1t

Y21 Y22 Y23 Y24 Y25 Y26 … Y2t

Y31 Y32 Y33 Y34 Y35 Y36 … Y3t

Y41 Y42 Y43 Y44 Y45 Y46 … Y4t

. . . . . . … .

. . . . . . … .

. . . . . . … .Yr1 Yr2 Yr3 Yr4 Yr5 Yr6 … Yrt_________________________________________________________________________________ __ __ __ __ __ __Y.1 Y.2 Y.3 Y.4 Y.5 Y.6 … Y.t

1 2 3 4 5 6 … t

Y•1, Y•2, …, are Column Means_ _

Y• • = Y• j /t = “GRAND MEAN”

(assuming same # data points in each column)

(otherwise, Y• • = mean of all the data)

j=1

t

MODEL: Yij = + j + ij

Y• • estimates

Y • j - Y • • estimatesj (= j – ) (for all j)

These estimates are based on Gauss’ (1796)

PRINCIPLE OF LEAST SQUARES

and on COMMON SENSE

MODEL: Yij = + j + ij

If you insert the estimates into the MODEL, (1) Yij = Y • • + (Y•j - Y • • ) + ij.

it follows that our estimate of ij is

(2) ij = Yij - Y•j

<

<

Then, Yij = Y• • + (Y• j - Y• • ) + ( Yij - Y• j)

or, (Yij - Y• • ) = (Y•j - Y• •) + (Yij - Y•j ) { { {(3)

TOTAL

VARIABILITY

in Y

=Variability

in Y

associated

with X

Variability

in Y

associated

with all other

factors

+

If you square both sides of (3), and double sum both sides (over i and j), you get, [after some unpleasant algebra, but lots of terms

which “cancel”]

(Yij - Y• • )2 = R • (Y•j - Y• •)

2 + (Yij - Y•j)2t r

j=1 i=1 { { {j=1

t t r

j=1 i=1

TSSTOTAL SUM OF

SQUARES

=

=

SSBC SUM OF

SQUARES BETWEEN COLUMNS

+

+

SSW (SSE)SUM OF SQUARES WITHIN COLUMNS( ( (

( ((

ANOVA TABLES V SS DF Mean

square (M.S.)

Among treatment (among columns)

Within Columns (due to error)

SSAc t - 1 MSACSSACt- 1

SSWc (r - 1) •t

SSWc(r-1)•t = MSW

=

TOTAL TSS tr -1

Hypothesis,HO: 1 = 2 = • • • c = 0

HI: not all j = 0

Or

HO: 1 = 2 = • • • • c

HI: not all j are EQUAL

(All column means are equal)

The probability Law of MSB

C MSWc

= “Fcalc” , is

The F - distribution with (t-1, (r-1)t)degrees of freedom

Assuming HO true.

Table Value

Example: Reed ManufacturingExample: Reed ManufacturingFaculty of Agriculture, GMU would like

to know if the teaching quality of xperimental design is similar among

classes .

A simple random sample of 5 student from 3 classes was taken and the grade

of experimental design was collected

Sample DataSample Data

ObservationObservation Advance Broadway Advance Broadway CindyCindy 11 0 066 0909 0404 22 0808 0404 1010 33 1 100 0707 1010 44 0606 0505 0505 55 1010 0505 0606

Sample MeanSample Mean 08 08 0606 0077 Sample VarianceSample Variance 04 04 0404

08 08

Example: Example: Grade of experimental designGrade of experimental design

HypothesesHypotheses

HH00:: 11==22==33

HHaa: Not all the means are equal: Not all the means are equalwhere:where: 1 1 = = Advance class Advance class 2 2 = = Broadway classBroadway class3 3 = = Cindy class Cindy class

Example: Example: Experimental DesignExperimental Design

Mean Square Due to TreatmentsMean Square Due to Treatments Since the sample sizes are all equalSince the sample sizes are all equal

μμ= (= (88 + + 66 + 7)/3 = + 7)/3 = 77 SSTR = 5(SSTR = 5(88 - - 77))22 + 5( + 5(66 - - 77))22 + 5(7 - + 5(7 - 77))22 = =

1010 MSTR = MSTR = 110/(3 - 1) = 0/(3 - 1) = 55

Mean Square Due to ErrorMean Square Due to ErrorSSE = 4(SSE = 4(44) + 4() + 4(44) + 4() + 4(88) = ) = 6464MSE = MSE = 6464/(15 - 3) =/(15 - 3) = 5.33 5.33

=


FF - Test - TestIf If HH00 is true, the ratio MSTR/MSE is true, the ratio MSTR/MSE should be should be near 1 because both MSTR and MSE are near 1 because both MSTR and MSE are estimatingestimating 22. . If If HHaa is true, the ratio should be is true, the ratio should be significantly larger than 1 because significantly larger than 1 because MSTR tends to overestimateMSTR tends to overestimate 22..



Rejection RuleRejection RuleUsing test statistic: Reject Using test statistic: Reject HH00 if if FF > 3.89 > 3.89Using Using pp-value-value : Reject : Reject HH00 if if pp-value -value < .05< .05

where where FF.05.05 = 3.89 is based on an = 3.89 is based on an FF distribution with 2 numerator degrees of distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom and 12 denominator degrees of freedomfreedom


Test StatisticTest StatisticFF = MSTR/MSE = = MSTR/MSE = 5.005.00//55..3333 = = 0.9380.938

ConclusionConclusionFF = =0.9380.938 << FF.05.05 = 3.89, so we = 3.89, so we accept accept HH00. . There is no significant different quality There is no significant different quality among experimental design classesamong experimental design classes

ANOVA TableANOVA Table

Source of Source of Sum of Degrees of Sum of Degrees of MeanMean Variation Variation Squares Freedom Squares Freedom Square FSquare Fcalc.calc. Among classesAmong classes 10 10 2 2 5.005.00 0.938 0.938 Within classesWithin classes 6464 12 12 5.335.33 Total Total 7744 1414


Step 1Step 1 Select the Select the ToolsTools pull-down menu pull-down menu Step 2Step 2 Choose the Choose the Data AnalysisData Analysis option option Step 3Step 3 Choose Choose Anova: Single FactorAnova: Single Factor

from the list of Analysis Toolsfrom the list of Analysis Tools

Using Excel’s Anova: Using Excel’s Anova: Single Factor Tool Single Factor Tool

Step 4Step 4 When the Anova: Single Factor dialog box When the Anova: Single Factor dialog box appears:appears: Enter B1:D6 in the Enter B1:D6 in the Input RangeInput Range box box Select Grouped By Select Grouped By ColumnsColumns Select Select Labels in First RowLabels in First Row Enter .05 in the Enter .05 in the AlphaAlpha box box Select Select Output RangeOutput Range Enter A8 (your choice) in the Enter A8 (your choice) in the Output RangeOutput Range boxbox Click Click OKOK

Using Excel’s Anova: Using Excel’s Anova: Single Factor ToolSingle Factor Tool

Value Worksheet (top portion)Value Worksheet (top portion)1 Observation Advance Broadway Cindy2 1 6 9 4 3 2 8 4 104 3 10 7 105 4 6 5 5 6 5 10 5 6 7

Using Excel’s Anova:Using Excel’s Anova: Single Factor Tool Single Factor Tool

Value Worksheet (bottom portion)Value Worksheet (bottom portion)


10 SUMMARY11 Groups Count Sum Average Variance12 Advance 5 40 8 413 Broadway 5 30 6 414 Cindy 5 35 7 8151617 ANOVA18 Source of Variation SS df MS F P-value F crit19 Among Groups 10 2 5,000 0,9375 0,00331 3,8852920 Within Groups 64 12 5,3332122 Total 74 1423 24

Using the Using the pp-Value-ValueThe value worksheet shows that the The value worksheet shows that the pp--

value is .00331value is .00331The rejection rule is “The rejection rule is “Reject Reject HH00 if if pp--

value < .05”value < .05”Thus, we reject Thus, we reject HH00 because the because the pp-value -value

= .00331 <= .00331 < = .05= .05We conclude that the We conclude that the quality of among quality of among

experimental design classes is similarexperimental design classes is similar


inferential statistics inferential statistics: the part of statistics that allows researchers to...

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