Fisher’s least significant difference (LSD)

Fisher's least significant difference (LSD)

Procedure:

1. Perform overall test of H!: ." oe .# oe ? oe .> vs. Ha: ." ? .# ? ? ? .> 2. If outcome is "do not reject H!, then stop. Otherwise continue to #3.

3. Perform desired hypothesis tests of (preplanned) paired comparisons using contrast tests:

H!: .3 oe .4 Ha: .3 ? .4 (two sided)

The LSD aspect comes from the rejection region for the test statistic

> oe s6 34 + 0 oe

+C 3 + +C 4

?Zs Ss6 34<

?=W#

S

1 83

/

1 84

<

One rejects H! if k>k >!?# (with 8T + > df). This is the same as: reject H! if

?+C 3 ++C 4?

>!?#

?=W# OE

1 83

/

1 84 (

When the sample sizes are equal (each 83 oe 8), the quantity on the right hand side becomes >!?# ?=W# a2?8b , which is the original meaning of the "least significant difference".

In SAS:

? PROC GLM; CLASS TRT; MODEL Y=TRT; MEANS TRT / LSD; (can use T instead of LSD

for same results)

"Story" is: the overall AOV test protects the EER for the mean comparisons.

? story based on simulations from early '70s ? it is not really true for many means ? still recommended statistical practice, when

EER is not taken too seriously ? confidence intervals instead of hypothesis

tests are calculated with the usual methods for contrasts:

+C 3

+ +C 4

,,

>!?#

?=W#

OE

1 83

/

1 84 (

Tukey's procedure

Tukey's procedure is based on the studentized range distribution

\", \#, ..., \8 a normal random sample; W is an independent estimate of 5, then

\max + \min W

?

studentized range

The procedure assume equal sample sizes ( oe 8) from each population. Then, under the hypothesis that the group means are the same ( oe .), the ]+3 are a random sample from a normala., 5#?8b distribution, and (a property of the normal) the sample means are independent of WW# . Thus, under the hypothesis that the means are the same,

]+3(max) + ]+4(min)

?aWW#?8b

has a studentized range distribution with (8T + >) df.

This statistic is based on the the maximum difference between the means. EER protection is accomplished by using the critical value for this statistic for all the mean comparisons.

Hypotheses

H!: .3 + .4 oe 0 Ha: .3 + .4 ? 0

Test statistic

; oe ?+C 3++C 4?

?a=W# ?8b

Rejection region

reject H! if ; ;! where ;! is the percentile from the studentized range distribution corresponding to > treatment means and 8T + > df. Selected percentiles appear in the text, Table 10, p. 1115-1116.

Text writes rejection region as:

reject H! if ?+C 3 ++C 4? ;!?a=W# ?8b a oe [ b

Simultaneous CIs (family-wide confidence is 100a1 + !b%) for the differences between means:

+C 3 ++C 4 ,, ;!?a=W# ?8b

Modification of Tukey procedure for unequal sample sizes

Tukey-Kramer

procedure

uses

;!

?

1 2

=W#

S

1 83

/

1 84

<

instead

of ;!?a=W# ?8b in all the formulas. The procedure has not been proven to protect the EER, but it has performed well

in simulations.

In SAS, the Tukey (or Tukey-Kramer, when sample sizes are unequal) procedure is invoked by

? PROC GLM; CLASS TRT; MODEL Y=TRT; MEANS TRT / TUKEY;

Tukey indeed protects EER, and Tukey-Kramer seems to do so. Comments:

? The tests and CIs are conservative ? Not necessary to protect with overall AOV

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download