Hypothesis tes+ng involving one or two samples: t test!!!!

p value quantifies random error

Hypothesis  tes+ng  involving  one   or    two  samples:  t  test!!!!   Laurens  Holmes,  Jr.   Board  Cer+fied  Public  Health,  FACE  

One  sample/two  samples   •  One  sample     •  Hypothesis  tes+ng  is  developed  and  applied  to  one-­‐ sample  problems  of  sta+s+cal  inference   •  Specified  about  a  single  sample  

•  Two  Sample   •  Two  sample  problem   •  Compares  two  different  distribu+ons  

Terms:  Standard  Error/Standard   Devia+on   •  •  •  •  •  •  •  •  • 

SE  or  SEM  refers  to  the  quan+ta+ve  measure  of  the  variability    of  sample  means   obtained  from  repeated  random  samples  of  size  n  drawn  from  the  same   popula+on.   Rela+onships:   SE  is  directly  propor+onal  to  inverse  of  the  (a)  square  root  of  the  sample  size  n  (1/ √n),  (b)  popula+on  SD  σ  of  individual  observa+ons.   The  larger  n  (sample  size),  the  more  precise  the  es+mate  of  μ.   Sample  precision  is  affected  by  variance  the  square  root  of  the  variance.   Implica+ons:   Sample  size  is  essen+al  in  assessing  the  precision  of  our  es+mate  X  of  the  unknown   popula+on  mean  μ.    SE  =  SD  /  √n   Vigne\e:  Suppose    a  mother  wishes  to  determine  exact  temp  of  her  son  or   daughter  for  the  purpose  a  fever-­‐induced  seizure.  If  there    is  a  theory  that  seizure   occurs    temp  eleva+on  by  an  amount  0.5  to  1.0  degrees  Fahrenheit.  Using  these   temps  (10  days),  es+mate  mean.  How  precise  is  this  es+mate?  

Vigne\e  :  Body  temperature  and   childhood  seizures   Day  

1  

Temp   97.2  

2  

3  

4  

5  

6  

7  

8  

9  

10  

96.8  

97.4  

97.4  

97.3  

97.0  

97.1  

97.3  

97.3  

97.2  

What  is   the  mean?  

What  is   the  SEM?  

Can  θΔ  be   used  to   seizure  day?  

Hypothesis  Tes+ng   •  Ra+onale?  Hypothesis  tes+ng  provides  an  objec+ve  framework  for   making  decision  using  probabilis+c    methods  rather  than  relying  on   subjec+ve  impression.   •  Provides    a  uniform    decision-­‐making  criterion  that  is  consistent  for  other   clinician  and  methodologists.    

•  •  •  •  • 

What  are  parameters?   What  ideas  do  we  have?   Do  our  data  conform  to  the  ideas?   Vigne\e:   Suppose  it  is  known  that  the  average  cholesterol  level  in  healthy  US   children  is175mg/dl.  There  were  a  group  of  men  who  died  from   coronary  heart  disease  (CAD)  within  the  past  year.  Suppose  too  that   we  have  the  cholesterol  level  of  their  offspring.     •  What  hypothesis  may  be  tested  in  this  situa+on?  

Terms:  Types  I  and  II  Errors   •  Hypothesis  tes+ng  a\empt  to  draw  conclusion  about  the  real  world   situa+on  based  on  the  results  of  the  sta+s+cal  test   •  There  are  4  possible  outcomes  in  hypothesis  tes+ng   –  –  –  – 

Ho  is  true  and  Ho  is  accepted  (Correct  decision)   H1  is  true  and  Ho  is  accepted  (Type  I  error)   Ho  is  true  and  Ho  is  rejected  (Correct  decision)   H1  is  true  and  Ho  is  rejected  (Type  II  error)  

•  Type  I  error  (α):  Probability  of  rejec+ng  the  null  hypothesis  when  indeed   the  null  is  true   •  Type  II  error  (β):  Probability  of  accep+ng  the  null  hypothesis  when  indeed   the  alterna+ve  is  true.     •  Significance  level    (pre-­‐established,  conven+onally  at  5%)    is  used  to   determine  the  probability  of  making  a  type  I  error.   •  With  α  <  0.05  we  reject  the  null  hypothesis  of  no  difference  and  incline  towards  the   alternate  hypothesis  of  difference  –  “sta+s+cally  significant”,  and  implies  that  the   difference  is  not  due  to  random  error.        

Hypothesis  Tes+ng   •  What  is  the  rela+ve  probabili+es  of  obtaining  sample  data   under  testable  hypotheses?  

–  Hypotheses  are  mutually  exclusive  (both  cannot  be  correct)  and   all  inclusive  (one  must  be  true)    

•  Statement:       •  Ho  –  The  average  cholesterol  of  these  children  is  175mg/dl   –  States  that  any  observed  difference  is  caused  by  random  error  

•  H1  –  The  average  cholesterol  level  of  these  children  is  >   175mg/dl    

–  States  that  the  observed  difference  is  caused  by  a  systema+c   difference  between  groups  

Vigne\e   •  Suppose  we  obtain  birth  weights  from  100   consecu+ve,  full-­‐term,  live-­‐born  deliveries  from  the   maternity  ward  of    no-­‐name  hospital    in  a  low-­‐SES  area.   The  average  birth  weight,  based  on  na+onwide  survey   of  millions  of  deliveries  is  120oz,  and  the  mean  of  the   birth  weight  of  the  100  deliveries  is  115  oz,  with  a   sample  SD  of  24  oz.     •   Is  the  underlying  mean  birth  weight  from  the  no-­‐name   hospital  lower  than  the  US  na+onal  average?       •  Hints  –  Assump+on:  The  100  birth  weights  from  this  hospital  come   from  an  underlying    normal  distribu+on  with  unknown  mean  μ.    

   

Vigne\e:  Inference  when  the  popula+on  weight  is  known

•  Suppose  the  average  weight  of  children  0-­‐18   years  in  the  Delaware  (DE)  is  40kg.     •  Are  the    66  cerebral  palsy    children  who   underwent  spinal  fusion  for  curve  deformi+es   correc+on  different  in  weight  from  the  healthy   children  in  DE?         •  Explain  your  answer,  and  provide  the  data.    

Normality  Test  

Normality  Test:  skewness  /  kurtosis    

SPSS  Output   •  What  is  the  rela+ve  probabili+es  of  obtaining  the   sample  data  under  the  null  and  alterna+ve  hypothesis?   •  SPSS  output    

•  SPSS  Process:  Click  on  “analyses”,  then  “Compare   means”,  then  “One-­‐sample  T  test”  ,  select  the  variable   and  enter  into    the  “test  variable  box”,  enter  by  typing     the  known  popula+on  mean  in  the  window,  “ Test   value”.  

One/Single  Sample  t  test    

One  sample  t  test  Output  interpreta+on    

SPSS  Output  Interpreta+on   •  How  do  you  write  the  result?   •  A  single-­‐sample  t  test  compared  the  mean  of  “X”  to  a   popula+on  mean  of    “Y”  (state  the  value).  A  significant   difference  was  found,  (t(df)=  (value  of  t),  p  <  0.0001  or   the  exact  p  value.     •  The  sample  mean  of    (X..value)  (SD=…value)  was   significantly  <  than  the  popula+on  mean.   •  Meaning   –  Significant-­‐  sample  mean  is  not  equivalent  to  popula+on   mean   –  Non-­‐significant  –  Not  a  significant  difference  between  the   mean,  does  not  mean  that  the  means  are  equal.    

What  is  p  –  value?   •  The  alpha  level  at  which  we  would  be  indifferent   between  accep+ng  or  rejec+ng  the  null   hypothesis,  given  the  sample  data.   •  The  alpha  level  at  which  the  given  value  of  test   sta+s+c  such  as  t  would  be  on  the  borderline   between  the  acceptance  and  rejec+on  regions.   •  A  p  value  is  the  probability  under  the  null   hypothesis  of  obtaining  a  test  sta+s+c  as  extreme   as  or  more  extreme  than  the  observed  test   sta+s+c.   •         

Independent/Two  samples  t  Test   Compares  the  means  of  two  samples   that  are  normally  distributed  from   randomly  assigned  groups  

Two  sample  or  Independent  sample  T-­‐   test   •  Purpose   •  To  compare  the  underlying  parameters  of  two  different   popula+ons,  neither  of  whose  values  is  assumed  known.   •  Independent  sample:  Samples  are  independent  when  the   data  points  in  one  sample  are  unrelated  to  the  data  points   in  the  second  sample   •  Two  completely  different  group  or  dis+nct  popula+on  are   compared.     •  Design:   –  Longitudinal  study   –  Cross-­‐sec+onal  

Two  sample  t  test   •  Assump+ons   •  Normal  distribu+on  –  dependent  must  be  measured  in   interval  or  ra+o  scale   •  Independence  –  observa+ons  are  independent  implies   informa+on  about  one  is  unrelated  to  another.     •  Equal  variance  

Two  sample  t  sta+s+c   •  t  =  (x1-­‐x2)  /  σ√1/n1  +  1/n2   •  Assuming  that  the  underlying  variance  in  the  two   groups  are  the  same  σ1=σ2=  σ.  The  mean  and  the   variances  in  the  two  samples  are  denoted  by    x1,   x2  s12,  s22,  respec+vely.  X1  is  normally   distributed  with  mean  μ1  and  variance  σ  2/n1,   and  X2  is  normally  distributed  with  mean  μ2  and   variance  σ  2/n2.   •  Since  the    samples  are  independent,  x1-­‐x2  is   normally  distributed  with  μ1-­‐  μ2,  and  variance  σ2 (1/n1  +  1/n2).   •  Equal  variance:  t  =  x1-­‐x2  /  SD√1/n1  +  1/n2        

Vigne\e   •  Suppose  the  cholesterol  levels  of  ten  children  (2-­‐14   years)  of  whose  fathers  died  from  CAD  are:  165,  170,   156,  171,  178,  179,  171,  181,  169  and  181  mg/dl;  and   those  of  ten  children  whose  fathers  are  alive  and  have   no  CAD  are:  160,  165,  145,  146,  152,  143,  140,  139,   160,  and  141  mg/dl.   •   Is  the  underlying  variance  equal?    Equality  of  spread!!   •  What  is  the  ra+o  of  the  variance?    

•  Are  these  samples  independent?     •  Why?  

•   Is  there  a  mean  difference  in  cholesterol  level   comparing  the  two  samples?          

Independent  Sample  T  test  procedure   Perform  Normality  test  

Yes   Perform  test  for  equality  of  two   variances   No  

Perform  t  test  with  unequal   variance  

Yes   Perform  t  test  with  equal  variance  

Vigne\e   •  The  dataset  on  Deep  wound  infec+on  following   spinal  fusion  in  children  with  CP  provided   informa+on  hematocrit,  temperature,  es+mated   blood  loss,  and  parked  red  blood  cells.  Suppose  a   case-­‐control  study  was  designed,  and  the  inves+gator   is  interested  in  knowing  whether  or  not  the  cases   significantly  differed  from  the  controls  with  respect   to  temperature.     •  What  test  sta+s+c  is  appropriate  for  this  inference?  What  are  the   assump+ons?  

•  Using  SPSS,  perform  the  specific  test  and  interpret   the  results.      

Independent/two  Samples  t  test  

SPSS  Output  

SPSS  Output  Interpreta+on   •  There  are  two  components  to  the  interpreta+on   •  Group  sta+s+cs:   •  Provides  the  basic  descrip+ve  sta+s+cs  for  the  dependent  variables  for   each  value  of  the  independent  variable  (two  discrete  levels,  0,1)   •  The  mean  age  at  surgery  with  the  CP  children  who  were  diagnosed  with   deep  wound  infec+on  was  13.9  years,  SD=3.1,  and  for  the  control  or  non-­‐ cases,.  Mean  =  14.5  years,  SD,  3.5.     •  Levene’s  Test  for  equality  of  variance:    This  test  indicates  that  the  variances   for  the  cases  compared  to  the  control  do  not  differ  significantly,  F=  0.38,  p   =  0.54.   •   t  test  answer:  T  test  assumes  an  equality  of  means  as  the  null  hypothesis.   The  last  three  columns  on  the  independent  sample  table  provide  the  t,  df   and  significance  (p  value).  There  was  no  significant  difference  between  the   mean  age  of  the  two  groups,  (t(  64)=0.64,  p  =  0.52.  

Paired/Dependent  Sample  T  test   Paired  sample  design-­‐  when  each  data   point  of  the  first  sample  is  matched  and   is  related  to  a  unique  data  point  of  the   second  sample  

Paired  t  test  formula  

Paired,  correlated,  matched,   dependent  sample  t  test  

SPSS  Output  

Paired    t  test  Assump+ons   •  Assump+ons   •  Both  variables  (two  measurements  of  same   subject)    (pre  and  post)  are  normally   distributed   •  Are  measured  on  interval  or  ra+o  scales   •  If  different  scales  are  used,  scores  must  be   converted  to  z-­‐score  prior  to  t  test  

Paired    t  test  Interpreta+on   •  Three  components  of  the  output:   •  Descrip+ve  sta+s+cs:   •  Preopera+ve  mean  (  77.03  degrees),  SD  (20.0)  and  SE  of  the  66   children  with  CP.   •  Postopera+ve  mean  (26.6)  thoracic  curve  angle,  SD  (15.2)  and  SE   of  the  66  children  with  CP    

•  Pearson  correla+on  coefficient  –  r  =  0.6,  p  <  0.0001.   •  Paired  Differences:  Mean  (50.4  degrees),  SD,  SE  t   value  (23.5),  df  (59),  and  sig  (p  value),  <  0.0001.