[R] The qurey about kolmogorov-smirnov test & adding the trendline to graph

[priti desai]

I am hereby forwarding the data & method use to calculate the
Kolmogorov-Smirnov goodness of fit test made manually by me in R
launguage which deffers with the actual inbuilt formula as  shown below.
Further I have plot the graph in R. In that graph how to add trendline
(i.e. straight line passing through maximum points in plot) to a Plot.


R script is as follows please run this script to see the output in R.

========================= start ========================= 

# The data is as follows

data <- c( 0.01,  0.02, 0.04, 0.13,  0.17 , 0.19 , 0.21 , 0.27 , 0.27 ,
0.28,  0.29,  0.37,
           0.41,  0.49,  0.51,  0.52,  0.54,  0.57,  0.62,  0.63,  0.68,
0.73,  0.74, 0.79,
           0.81,  0.81,  0.82,  0.86,  0.94,  0.96,  1.02,  1.10,  1.10,
1.20,  1.29,  1.36,
           1.40,  1.41,  1.44,  1.45,  1.62,  1.67,  1.69,  1.78,  1.82,
2.11,  2.13,  2.14,
           2.24,  2.29,  2.34, 2.40,  2.46,  2.70,  2.83,  2.98,  3.00,
3.30,  3.53,  3.70,
           3.86,  3.90,  3.91,  3.98,  5.01,  5.23,  6.05,  6.12, 10.41,
10.73)



# K-S goodness of fit test (actual calculated)
print('K-S goodness of fit test')

# Ho:sample come from postulated prob distribution  vs   H1: Not Ho
# Reject Ho if o > 0.162312045 o.w. don't reject Ho at 5% los.
# Reject Ho if o > 0.194583 o.w. don't reject Ho at 1% los.
# In this case, we don't Reject Ho at 5% &1% los.

average       <- mean(data)  
print('average:')
# estimate the parameter
print('Estimation of parameter')

lambda        <- (1/average)
print(lambda)


e             <- c(1:70)
k             <- c((e-1)/70)
Fx            <- c(1 - exp(-lambda*data))
g             <- sort(Fx)
l             <- c(g-k)
m             <- c(e/70)
n             <- c(m-g)                        
o             <- max(l,n)
print('k-s stat:')
print(o)

#  K-S goodness of fit test (R inbuilt formula)


 ks.test(x= data,"pexp", alternative = c("g"),exact = NULL)



# P-P plot

 
e             <- c(1:70)
f             <- c((e-.5)/70)
Fx            <- c(1 - exp(-lambda*data))
g             <- sort(Fx)
plot(f,g)

========================= end ========================= 

The results are as follows:
  The K-S test calculated manualy giving the result as follows,
 
Ho:sample come from postulated prob distribution  vs   H1: Not Ho
Ks statistic = D = 0.04391726

 Reject Ho if o > 0.162312045 o.w. don't reject Ho at 5% los.
 Reject Ho if o > 0.194583 o.w. don't reject Ho at 1% los.
 In this case, we don't Reject Ho at 5% &1% los.

 (by inbuilt formula the output is as follows,)

 One-sample Kolmogorov-Smirnov test

data:  data 
D^+ = 0.0088, p-value = 0.9893
alternative hypothesis: greater 

Warning message:
cannot compute correct p-values with ties in: ks.test(x = data, "pexp",
alternative = c("g"), exact = NULL) 




It is requested to clarify & confirm formula derirved so as to enable 
me to cross check my calculations made manually. Further please convey 
as how to interprete the results. 
Awaiting your positive reply.

 
Regards,
Priti